{"id":96,"date":"2026-08-07T11:24:20","date_gmt":"2026-08-07T11:24:20","guid":{"rendered":"https:\/\/sqoolpapers.co.za\/notes\/patterns-sequences-and-series-g12-maths-t4p1-test-3\/"},"modified":"2026-08-22T16:28:51","modified_gmt":"2026-08-22T16:28:51","slug":"patterns-sequences-and-series-g12-maths-t4p1","status":"publish","type":"post","link":"https:\/\/repstaq.com\/study-guide-docs\/patterns-sequences-and-series-g12-maths-t4p1\/","title":{"rendered":"Patterns, Sequences and Series-G12-Maths-T4P1"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\">Patterns, Sequences and Series<\/h2>\n\n\n\n<h2 class=\"wp-block-heading\">Number patterns<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A number pattern is an ordered list of numbers that follows a rule. Each number is called a term. The position of a term is represented by n, while the term in that position is written as T_n.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">To identify a pattern:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Calculate the first differences between consecutive terms.<\/li>\n\n\n\n<li>If these are constant, the pattern is arithmetic.<\/li>\n\n\n\n<li>If each term is multiplied by the same number, the pattern is geometric.<\/li>\n\n\n\n<li>If the second differences are constant, the pattern is quadratic.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Example<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Consider the pattern:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mn>3<\/mn><mo separator=\"true\">;<\/mo><mtext>&nbsp;<\/mtext><mn>7<\/mn><mo separator=\"true\">;<\/mo><mtext>&nbsp;<\/mtext><mn>11<\/mn><mo separator=\"true\">;<\/mo><mtext>&nbsp;<\/mtext><mn>15<\/mn><mo separator=\"true\">;<\/mo><mo>\u2026<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">3;\\ 7;\\ 11;\\ 15;\\ldots<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">The first differences are:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mn>7<\/mn><mo>\u2212<\/mo><mn>3<\/mn><mo>=<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">7-3=4<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mn>11<\/mn><mo>\u2212<\/mo><mn>7<\/mn><mo>=<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">11-7=4<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mn>15<\/mn><mo>\u2212<\/mo><mn>11<\/mn><mo>=<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">15-11=4<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore, the pattern is arithmetic with a common difference of 4.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Remember:<\/strong> A sequence is a list of terms, while a series is the sum of the terms.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Arithmetic sequences<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">In an arithmetic sequence, the difference between consecutive terms is constant. This value is called the common difference, d.<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mi>d<\/mi><mo>=<\/mo><msub><mi>T<\/mi><mrow><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo>\u2212<\/mo><msub><mi>T<\/mi><mi>n<\/mi><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">d=T_{n+1}-T_n<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">The general term is:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><msub><mi>T<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mi>a<\/mi><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mi>d<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">T_n=a+(n-1)d<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Here, a is the first term, d is the common difference and n is the term number.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Example<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Determine the 20th term of:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mn>3<\/mn><mo separator=\"true\">;<\/mo><mtext>&nbsp;<\/mtext><mn>7<\/mn><mo separator=\"true\">;<\/mo><mtext>&nbsp;<\/mtext><mn>11<\/mn><mo separator=\"true\">;<\/mo><mo>\u2026<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">3;\\ 7;\\ 11;\\ldots<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a=3<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mi>d<\/mi><mo>=<\/mo><mn>7<\/mn><mo>\u2212<\/mo><mn>3<\/mn><mo>=<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">d=7-3=4<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><msub><mi>T<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mi>a<\/mi><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mi>d<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">T_n=a+(n-1)d<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><msub><mi>T<\/mi><mn>20<\/mn><\/msub><mo>=<\/mo><mn>3<\/mn><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>20<\/mn><mo>\u2212<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>4<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">T_{20}=3+(20-1)(4)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><msub><mi>T<\/mi><mn>20<\/mn><\/msub><mo>=<\/mo><mn>3<\/mn><mo>+<\/mo><mn>76<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">T_{20}=3+76<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><msub><mi>T<\/mi><mn>20<\/mn><\/msub><mo>=<\/mo><mn>79<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">T_{20}=79<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">An important property of an arithmetic sequence is that terms equally far from a central term have a constant difference determined by their positions. For example, if the common difference is 4:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><msub><mi>T<\/mi><mn>14<\/mn><\/msub><mo>\u2212<\/mo><msub><mi>T<\/mi><mn>6<\/mn><\/msub><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>14<\/mn><mo>\u2212<\/mo><mn>6<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>4<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>32<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">T_{14}-T_6=(14-6)(4)=32<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">This is a useful shortcut in examination questions.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/sqooltutors.co.za\/sign-up-b\/\"><img decoding=\"async\" src=\"https:\/\/lmxddlwowqlsyucleuyv.supabase.co\/storage\/v1\/object\/public\/pdf_ads\/batch%201\/Doc%20Image%20Aug%2022,%202026,%2004_21_42%20PM%20(1).png\" alt=\"\"\/><\/a><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Geometric sequences<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">In a geometric sequence, each term is obtained by multiplying the previous term by a constant called the common ratio, r.<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mi>r<\/mi><mo>=<\/mo><mfrac><msub><mi>T<\/mi><mrow><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/msub><msub><mi>T<\/mi><mi>n<\/mi><\/msub><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">r=\\frac{T_{n+1}}{T_n}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">The general term is:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><msub><mi>T<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mi>a<\/mi><msup><mi>r<\/mi><mrow><mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">T_n=ar^{n-1}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Example<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Determine the 25th term of:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mo>\u2212<\/mo><mn>8<\/mn><mo separator=\"true\">;<\/mo><mtext>&nbsp;<\/mtext><mn>4<\/mn><mo separator=\"true\">;<\/mo><mtext>&nbsp;<\/mtext><mo>\u2212<\/mo><mn>2<\/mn><mo separator=\"true\">;<\/mo><mtext>&nbsp;<\/mtext><mn>1<\/mn><mo separator=\"true\">;<\/mo><mo>\u2026<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">-8;\\ 4;\\ -2;\\ 1;\\ldots<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>8<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a=-8<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mi>r<\/mi><mo>=<\/mo><mfrac><mn>4<\/mn><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mn>8<\/mn><\/mrow><\/mfrac><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">r=\\frac{4}{-8}=-\\frac{1}{2}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><msub><mi>T<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mi>a<\/mi><msup><mi>r<\/mi><mrow><mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">T_n=ar^{n-1}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><msub><mi>T<\/mi><mn>25<\/mn><\/msub><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>8<\/mn><msup><mrow><mo fence=\"true\" form=\"prefix\">(<\/mo><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mn>24<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">T_{25}=-8\\left(-\\frac{1}{2}\\right)^{24}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><msub><mi>T<\/mi><mn>25<\/mn><\/msub><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>8<\/mn><msup><mrow><mo fence=\"true\" form=\"prefix\">(<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mn>24<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">T_{25}=-8\\left(\\frac{1}{2}\\right)^{24}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><msub><mi>T<\/mi><mn>25<\/mn><\/msub><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mfrac><mn>1<\/mn><msup><mn>2<\/mn><mn>21<\/mn><\/msup><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">T_{25}=-\\frac{1}{2^{21}}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">This example shows why the sign of the exponent must be handled carefully when the common ratio is negative.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Quadratic number patterns<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A quadratic number pattern has constant second differences. Its general term has the form:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><msub><mi>T<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mi>a<\/mi><msup><mi>n<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mi>b<\/mi><mi>n<\/mi><mo>+<\/mo><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">T_n=an^2+bn+c<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">The constant second difference equals 2a.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Example<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Determine the general term of:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mn>3<\/mn><mo separator=\"true\">;<\/mo><mtext>&nbsp;<\/mtext><mn>8<\/mn><mo separator=\"true\">;<\/mo><mtext>&nbsp;<\/mtext><mn>15<\/mn><mo separator=\"true\">;<\/mo><mtext>&nbsp;<\/mtext><mn>24<\/mn><mo separator=\"true\">;<\/mo><mo>\u2026<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">3;\\ 8;\\ 15;\\ 24;\\ldots<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">First differences:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mn>5<\/mn><mo separator=\"true\">;<\/mo><mtext>&nbsp;<\/mtext><mn>7<\/mn><mo separator=\"true\">;<\/mo><mtext>&nbsp;<\/mtext><mn>9<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">5;\\ 7;\\ 9<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Second differences:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mn>2<\/mn><mo separator=\"true\">;<\/mo><mtext>&nbsp;<\/mtext><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2;\\ 2<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mn>2<\/mn><mi>a<\/mi><mo>=<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2a=2<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a=1<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Use the first two terms to determine b and c.<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><msub><mi>T<\/mi><mn>1<\/mn><\/msub><mo>=<\/mo><mn>1<\/mn><mo>+<\/mo><mi>b<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">T_1=1+b+c=3<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mi>b<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">b+c=2<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><msub><mi>T<\/mi><mn>2<\/mn><\/msub><mo>=<\/mo><mn>4<\/mn><mo>+<\/mo><mn>2<\/mn><mi>b<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>8<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">T_2=4+2b+c=8<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mn>2<\/mn><mi>b<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2b+c=4<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Subtract the equations:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mi>b<\/mi><mo>=<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">b=2<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">c=0<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><msub><mi>T<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><msup><mi>n<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>2<\/mn><mi>n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">T_n=n^2+2n<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">General term of a sequence<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The general term gives the value of any term directly without listing all the preceding terms.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For an arithmetic sequence:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><msub><mi>T<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mi>a<\/mi><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mi>d<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">T_n=a+(n-1)d<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">For a geometric sequence:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><msub><mi>T<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mi>a<\/mi><msup><mi>r<\/mi><mrow><mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">T_n=ar^{n-1}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">For a quadratic sequence:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><msub><mi>T<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mi>a<\/mi><msup><mi>n<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mi>b<\/mi><mi>n<\/mi><mo>+<\/mo><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">T_n=an^2+bn+c<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Example<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Determine which term of the sequence below is 467.<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mn>3<\/mn><mo separator=\"true\">;<\/mo><mtext>&nbsp;<\/mtext><mn>7<\/mn><mo separator=\"true\">;<\/mo><mtext>&nbsp;<\/mtext><mn>11<\/mn><mo separator=\"true\">;<\/mo><mo>\u2026<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">3;\\ 7;\\ 11;\\ldots<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a=3<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mi>d<\/mi><mo>=<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">d=4<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><msub><mi>T<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mi>a<\/mi><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mi>d<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">T_n=a+(n-1)d<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mn>467<\/mn><mo>=<\/mo><mn>3<\/mn><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>4<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">467=3+(n-1)(4)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mn>467<\/mn><mo>=<\/mo><mn>3<\/mn><mo>+<\/mo><mn>4<\/mn><mi>n<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">467=3+4n-4<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mn>467<\/mn><mo>=<\/mo><mn>4<\/mn><mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">467=4n-1<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mn>468<\/mn><mo>=<\/mo><mn>4<\/mn><mi>n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">468=4n<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mi>n<\/mi><mo>=<\/mo><mn>117<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">n=117<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore, 467 is the 117th term.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/sqooltutors.co.za\/sign-up-b\/\"><img decoding=\"async\" src=\"https:\/\/lmxddlwowqlsyucleuyv.supabase.co\/storage\/v1\/object\/public\/pdf_ads\/batch%201\/Doc%20Image%20Aug%2022,%202026,%2004_21_46%20PM%20(1).png\" alt=\"\"\/><\/a><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Arithmetic series<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">An arithmetic series is the sum of the terms of an arithmetic sequence.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The sum of the first n terms is:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><msub><mi>S<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mfrac><mi>n<\/mi><mn>2<\/mn><\/mfrac><mrow><mo fence=\"true\" form=\"prefix\">[<\/mo><mn>2<\/mn><mi>a<\/mi><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mi>d<\/mi><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">S_n=\\frac{n}{2}\\left[2a+(n-1)d\\right]<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Alternatively, if the last term is known:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><msub><mi>S<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mfrac><mi>n<\/mi><mn>2<\/mn><\/mfrac><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><msub><mi>T<\/mi><mi>n<\/mi><\/msub><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">S_n=\\frac{n}{2}(a+T_n)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Example<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Calculate the sum of the first 20 terms of:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mn>3<\/mn><mo>+<\/mo><mn>7<\/mn><mo>+<\/mo><mn>11<\/mn><mo>+<\/mo><mo>\u2026<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">3+7+11+\\ldots<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a=3<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mi>d<\/mi><mo>=<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">d=4<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mi>n<\/mi><mo>=<\/mo><mn>20<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">n=20<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><msub><mi>S<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mfrac><mi>n<\/mi><mn>2<\/mn><\/mfrac><mrow><mo fence=\"true\" form=\"prefix\">[<\/mo><mn>2<\/mn><mi>a<\/mi><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mi>d<\/mi><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">S_n=\\frac{n}{2}\\left[2a+(n-1)d\\right]<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><msub><mi>S<\/mi><mn>20<\/mn><\/msub><mo>=<\/mo><mfrac><mn>20<\/mn><mn>2<\/mn><\/mfrac><mrow><mo fence=\"true\" form=\"prefix\">[<\/mo><mn>2<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>3<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>20<\/mn><mo>\u2212<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>4<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">S_{20}=\\frac{20}{2}\\left[2(3)+(20-1)(4)\\right]<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><msub><mi>S<\/mi><mn>20<\/mn><\/msub><mo>=<\/mo><mn>10<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>6<\/mn><mo>+<\/mo><mn>76<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">S_{20}=10(6+76)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><msub><mi>S<\/mi><mn>20<\/mn><\/msub><mo>=<\/mo><mn>820<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">S_{20}=820<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Geometric series<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A geometric series is formed when the terms of a geometric sequence are added.<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mi>a<\/mi><mo>+<\/mo><mi>a<\/mi><mi>r<\/mi><mo>+<\/mo><mi>a<\/mi><msup><mi>r<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mi>a<\/mi><msup><mi>r<\/mi><mn>3<\/mn><\/msup><mo>+<\/mo><mo>\u2026<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">a+ar+ar^2+ar^3+\\ldots<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">The formula used depends on whether the series is finite or infinite.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Always determine a and r before selecting a formula. If the signs alternate, the common ratio is negative.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Finite geometric series<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A finite geometric series has a fixed number of terms. Its sum is:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><msub><mi>S<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mfrac><mrow><mi>a<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mi>r<\/mi><mi>n<\/mi><\/msup><mo>\u2212<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">)<\/mo><\/mrow><mrow><mi>r<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">S_n=\\frac{a(r^n-1)}{r-1}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Alternatively:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><msub><mi>S<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mfrac><mrow><mi>a<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><msup><mi>r<\/mi><mi>n<\/mi><\/msup><mo form=\"postfix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">)<\/mo><\/mrow><mrow><mn>1<\/mn><mo>\u2212<\/mo><mi>r<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">S_n=\\frac{a(1-r^n)}{1-r}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">These formulas apply when:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mi>r<\/mi><mo>\u2260<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">r\\ne 1<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Example<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Calculate the sum of the first five terms of:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mn>2<\/mn><mo>+<\/mo><mn>6<\/mn><mo>+<\/mo><mn>18<\/mn><mo>+<\/mo><mo>\u2026<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">2+6+18+\\ldots<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a=2<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mi>r<\/mi><mo>=<\/mo><mfrac><mn>6<\/mn><mn>2<\/mn><\/mfrac><mo>=<\/mo><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">r=\\frac{6}{2}=3<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><msub><mi>S<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mfrac><mrow><mi>a<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mi>r<\/mi><mi>n<\/mi><\/msup><mo>\u2212<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">)<\/mo><\/mrow><mrow><mi>r<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">S_n=\\frac{a(r^n-1)}{r-1}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><msub><mi>S<\/mi><mn>5<\/mn><\/msub><mo>=<\/mo><mfrac><mrow><mn>2<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mn>3<\/mn><mn>5<\/mn><\/msup><mo>\u2212<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">)<\/mo><\/mrow><mrow><mn>3<\/mn><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">S_5=\\frac{2(3^5-1)}{3-1}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><msub><mi>S<\/mi><mn>5<\/mn><\/msub><mo>=<\/mo><mfrac><mrow><mn>2<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>243<\/mn><mo>\u2212<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">)<\/mo><\/mrow><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">S_5=\\frac{2(243-1)}{2}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><msub><mi>S<\/mi><mn>5<\/mn><\/msub><mo>=<\/mo><mn>242<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">S_5=242<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Infinite geometric series<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">An infinite geometric series continues without ending:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mi>a<\/mi><mo>+<\/mo><mi>a<\/mi><mi>r<\/mi><mo>+<\/mo><mi>a<\/mi><msup><mi>r<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mi>a<\/mi><msup><mi>r<\/mi><mn>3<\/mn><\/msup><mo>+<\/mo><mo>\u2026<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">a+ar+ar^2+ar^3+\\ldots<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">A sum to infinity exists only if:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mo>\u2212<\/mo><mn>1<\/mn><mo>&lt;<\/mo><mi>r<\/mi><mo>&lt;<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">-1&lt;r&lt;1<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Equivalently:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">|<\/mo><mi>r<\/mi><mo form=\"postfix\" stretchy=\"false\">|<\/mo><mo>&lt;<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\lvert r\\rvert&lt;1<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">If the absolute value of r is greater than or equal to 1, the terms do not approach zero and the series has no sum to infinity.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Sum to infinity<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The sum to infinity of a convergent geometric series is:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><msub><mi>S<\/mi><mi>\u221e<\/mi><\/msub><mo>=<\/mo><mfrac><mi>a<\/mi><mrow><mn>1<\/mn><mo>\u2212<\/mo><mi>r<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">S_{\\infty}=\\frac{a}{1-r}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Example<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Calculate the sum to infinity of:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mo>\u2212<\/mo><mn>8<\/mn><mo>+<\/mo><mn>4<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mo>+<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mo>\u2026<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">-8+4-2+1-\\ldots<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>8<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a=-8<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mi>r<\/mi><mo>=<\/mo><mfrac><mn>4<\/mn><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mn>8<\/mn><\/mrow><\/mfrac><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">r=\\frac{4}{-8}=-\\frac{1}{2}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mrow><mo fence=\"true\" form=\"prefix\">|<\/mo><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><mo>&lt;<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\left|-\\frac{1}{2}\\right|&lt;1<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore, the series converges.<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><msub><mi>S<\/mi><mi>\u221e<\/mi><\/msub><mo>=<\/mo><mfrac><mi>a<\/mi><mrow><mn>1<\/mn><mo>\u2212<\/mo><mi>r<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">S_{\\infty}=\\frac{a}{1-r}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><msub><mi>S<\/mi><mi>\u221e<\/mi><\/msub><mo>=<\/mo><mfrac><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mn>8<\/mn><\/mrow><mrow><mn>1<\/mn><mo>\u2212<\/mo><mrow><mo fence=\"true\" form=\"prefix\">(<\/mo><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">S_{\\infty}=\\frac{-8}{1-\\left(-\\frac{1}{2}\\right)}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><msub><mi>S<\/mi><mi>\u221e<\/mi><\/msub><mo>=<\/mo><mfrac><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mn>8<\/mn><\/mrow><mfrac><mn>3<\/mn><mn>2<\/mn><\/mfrac><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">S_{\\infty}=\\frac{-8}{\\frac{3}{2}}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><msub><mi>S<\/mi><mi>\u221e<\/mi><\/msub><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mfrac><mn>16<\/mn><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">S_{\\infty}=-\\frac{16}{3}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">This substitution and answer agree with the expected examination method.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Sigma notation<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Sigma notation is a compact way of writing a series. The Greek capital letter sigma represents summation.<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mrow><munderover><mo movablelimits=\"false\">\u2211<\/mo><mrow><mi>k<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>n<\/mi><\/munderover><\/mrow><msub><mi>T<\/mi><mi>k<\/mi><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\\sum_{k=1}^{n}T_k<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">The lower value gives the starting value of the index, and the upper value gives the final value.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Example<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Expand:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mrow><munderover><mo movablelimits=\"false\">\u2211<\/mo><mrow><mi>k<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mn>4<\/mn><\/munderover><\/mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>k<\/mi><mo>+<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\sum_{k=1}^{4}(2k+1)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">[<\/mo><mn>2<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">]<\/mo><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">[<\/mo><mn>2<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>2<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">]<\/mo><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">[<\/mo><mn>2<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>3<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">]<\/mo><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">[<\/mo><mn>2<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>4<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">]<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">=[2(1)+1]+[2(2)+1]+[2(3)+1]+[2(4)+1]<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mo>=<\/mo><mn>3<\/mn><mo>+<\/mo><mn>5<\/mn><mo>+<\/mo><mn>7<\/mn><mo>+<\/mo><mn>9<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">=3+5+7+9<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mo>=<\/mo><mn>24<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">=24<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Useful sigma results include:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mrow><munderover><mo movablelimits=\"false\">\u2211<\/mo><mrow><mi>k<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>n<\/mi><\/munderover><\/mrow><mi>c<\/mi><mo>=<\/mo><mi>c<\/mi><mi>n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\sum_{k=1}^{n}c=cn<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mrow><munderover><mo movablelimits=\"false\">\u2211<\/mo><mrow><mi>k<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>n<\/mi><\/munderover><\/mrow><mi>k<\/mi><mo>=<\/mo><mfrac><mrow><mi>n<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">)<\/mo><\/mrow><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\sum_{k=1}^{n}k=\\frac{n(n+1)}{2}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mrow><munderover><mo movablelimits=\"false\">\u2211<\/mo><mrow><mi>k<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>n<\/mi><\/munderover><\/mrow><msup><mi>k<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mfrac><mrow><mi>n<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">)<\/mo><\/mrow><mn>6<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\sum_{k=1}^{n}k^2=\\frac{n(n+1)(2n+1)}{6}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Determining unknown terms and constants<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Unknown values are found by applying the defining property of the sequence or series.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Example: Unknown in an arithmetic sequence<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The following are consecutive terms of an arithmetic sequence:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo separator=\"true\">;<\/mo><mtext>&nbsp;<\/mtext><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo separator=\"true\">;<\/mo><mtext>&nbsp;<\/mtext><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x-3;\\ 2x+1;\\ 5x-7<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Consecutive differences must be equal.<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(2x+1)-(x-3)=(5x-7)-(2x+1)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><mo>=<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>8<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x+4=3x-8<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mn>12<\/mn><mo>=<\/mo><mn>2<\/mn><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">12=2x<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>6<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x=6<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">The terms are therefore:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mn>3<\/mn><mo separator=\"true\">;<\/mo><mtext>&nbsp;<\/mtext><mn>13<\/mn><mo separator=\"true\">;<\/mo><mtext>&nbsp;<\/mtext><mn>23<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">3;\\ 13;\\ 23<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Example: Unknown in a geometric sequence<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Suppose 2, k and 18 are consecutive terms. Equal ratios give:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mfrac><mi>k<\/mi><mn>2<\/mn><\/mfrac><mo>=<\/mo><mfrac><mn>18<\/mn><mi>k<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{k}{2}=\\frac{18}{k}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><msup><mi>k<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mn>36<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">k^2=36<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><semantics><mrow><mi>k<\/mi><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u00b1<\/mo><mn>6<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">k=\\pm 6<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Both values must be considered unless the question states that the terms are positive.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Exam Tip<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Write down the correct formula before substituting. Show the number of terms, common difference or common ratio, substitution, simplification and final answer. When solving for an index, reject negative or non-integer values because a term number must be a positive integer. Examination marking commonly awards separate marks for these stages and for selecting the valid solution.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Patterns, Sequences and Series Number patterns A number pattern is an ordered list of numbers that follows a rule. Each number is called a term. The position of a term is represented by n, while the term in that position is written as T_n. To identify a pattern: Example Consider the pattern: The first differences [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"_breakdance_hide_in_design_set":false,"_breakdance_tags":"","footnotes":""},"categories":[1],"tags":[],"class_list":["post-96","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"acf":{"document_name":"Patterns, Sequences and Series","grade":12,"subject":"Mathematics","term":4,"paper":1},"_links":{"self":[{"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/posts\/96","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/comments?post=96"}],"version-history":[{"count":3,"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/posts\/96\/revisions"}],"predecessor-version":[{"id":232,"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/posts\/96\/revisions\/232"}],"wp:attachment":[{"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/media?parent=96"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/categories?post=96"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/tags?post=96"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}