{"id":256,"date":"2026-09-09T06:15:30","date_gmt":"2026-09-09T06:15:30","guid":{"rendered":"https:\/\/repstaq.com\/study-guide-docs\/number-patterns-g11-maths-t4p1\/"},"modified":"2026-09-09T06:15:30","modified_gmt":"2026-09-09T06:15:30","slug":"number-patterns-g11-maths-t4p1","status":"publish","type":"post","link":"https:\/\/repstaq.com\/study-guide-docs\/number-patterns-g11-maths-t4p1\/","title":{"rendered":"Number patterns-G11-Maths-T4P1"},"content":{"rendered":"\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 called its term number. Number patterns may be represented numerically, algebraically, verbally or graphically.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">When studying a pattern, identify:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The first few terms.<\/li>\n<li>The differences between consecutive terms.<\/li>\n<li>Whether the differences are constant.<\/li>\n<li>A rule that connects the term number to the value of the term.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">Linear patterns<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A linear pattern increases or decreases by the same amount between consecutive terms. Therefore, its first differences are constant.<\/p>\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\"><mrow><mn>5<\/mn><mo>;<\/mo><mn>8<\/mn><mo>;<\/mo><mn>11<\/mn><mo>;<\/mo><mn>14<\/mn><mo>;<\/mo><mn>17<\/mn><mo>;<\/mo><mo>\u2026<\/mo><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Calculate the first differences:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mn>8<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mo>=<\/mo><mn>3<\/mn><\/mrow><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mn>11<\/mn><mo>\u2212<\/mo><mn>8<\/mn><mo>=<\/mo><mn>3<\/mn><\/mrow><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mn>14<\/mn><mo>\u2212<\/mo><mn>11<\/mn><mo>=<\/mo><mn>3<\/mn><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">The constant first difference is:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mi>d<\/mi><mo>=<\/mo><mn>3<\/mn><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">The pattern is linear because the first differences are constant.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">If the ordered pairs formed by the term numbers and term values are plotted on a Cartesian plane, the points of a linear pattern lie on a straight line. The common difference gives the gradient of this line.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A linear rule has the form:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msub><mi>T<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mi>d<\/mi><mi>n<\/mi><mo>+<\/mo><mi>c<\/mi><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">In this form, the common difference is the gradient and the constant term is the y-intercept of the extended graph.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Arithmetic sequences<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">An arithmetic sequence is a number pattern in which a constant number is added to or subtracted from each term to obtain the next term. This constant number is called the common difference.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The common difference is calculated using:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mi>d<\/mi><mo>=<\/mo><msub><mi>T<\/mi><mn>2<\/mn><\/msub><mo>\u2212<\/mo><msub><mi>T<\/mi><mn>1<\/mn><\/msub><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">It can also be checked using any two consecutive terms:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><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><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Example<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Consider the arithmetic sequence:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mn>20<\/mn><mo>;<\/mo><mn>16<\/mn><mo>;<\/mo><mn>12<\/mn><mo>;<\/mo><mn>8<\/mn><mo>;<\/mo><mn>4<\/mn><mo>;<\/mo><mo>\u2026<\/mo><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Calculate the common difference:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mi>d<\/mi><mo>=<\/mo><mn>16<\/mn><mo>\u2212<\/mo><mn>20<\/mn><\/mrow><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mi>d<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>4<\/mn><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">This is a decreasing arithmetic sequence because the common difference is negative.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">An arithmetic sequence can also be described recursively. A recursive rule states how to obtain each term from the previous term:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msub><mi>T<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><msub><mi>T<\/mi><mrow><mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo>+<\/mo><mi>d<\/mi><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">For the sequence above:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msub><mi>T<\/mi><mn>1<\/mn><\/msub><mo>=<\/mo><mn>20<\/mn><\/mrow><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msub><mi>T<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><msub><mi>T<\/mi><mrow><mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo>\u2212<\/mo><mn>4<\/mn><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">A recursive rule requires the previous term. A general term allows any term to be calculated directly.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">General term Tn<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The general term of an arithmetic sequence is:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msub><mi>T<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mi>a<\/mi><mo>+<\/mo><mo>(<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>)<\/mo><mi>d<\/mi><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">In this formula, the first term is represented by:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mi>a<\/mi><mo>=<\/mo><msub><mi>T<\/mi><mn>1<\/mn><\/msub><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">The term number is represented by:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mi>n<\/mi><mo>\u2208<\/mo><mi mathvariant=\"double-struck\">N<\/mi><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">The common difference is represented by:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mi>d<\/mi><mo>=<\/mo><msub><mi>T<\/mi><mn>2<\/mn><\/msub><mo>\u2212<\/mo><msub><mi>T<\/mi><mn>1<\/mn><\/msub><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Example<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Find the general term of:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mn>7<\/mn><mo>;<\/mo><mn>11<\/mn><mo>;<\/mo><mn>15<\/mn><mo>;<\/mo><mn>19<\/mn><mo>;<\/mo><mo>\u2026<\/mo><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Calculate the common difference:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mi>d<\/mi><mo>=<\/mo><mn>11<\/mn><mo>\u2212<\/mo><mn>7<\/mn><\/mrow><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mi>d<\/mi><mo>=<\/mo><mn>4<\/mn><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Identify the first term:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mi>a<\/mi><mo>=<\/mo><mn>7<\/mn><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Substitute into the general formula:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msub><mi>T<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mi>a<\/mi><mo>+<\/mo><mo>(<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>)<\/mo><mi>d<\/mi><\/mrow><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msub><mi>T<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mn>7<\/mn><mo>+<\/mo><mo>(<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>)<\/mo><mo>(<\/mo><mn>4<\/mn><mo>)<\/mo><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Simplify:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msub><mi>T<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mn>7<\/mn><mo>+<\/mo><mn>4<\/mn><mi>n<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msub><mi>T<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mn>4<\/mn><mi>n<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Exam Tip<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Always test the general term by substituting the first two term numbers.<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msub><mi>T<\/mi><mn>1<\/mn><\/msub><mo>=<\/mo><mn>4<\/mn><mo>(<\/mo><mn>1<\/mn><mo>)<\/mo><mo>+<\/mo><mn>3<\/mn><mo>=<\/mo><mn>7<\/mn><\/mrow><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msub><mi>T<\/mi><mn>2<\/mn><\/msub><mo>=<\/mo><mn>4<\/mn><mo>(<\/mo><mn>2<\/mn><mo>)<\/mo><mo>+<\/mo><mn>3<\/mn><mo>=<\/mo><mn>11<\/mn><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">The calculated values agree with the original sequence.<\/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 a general term containing a squared term. Its general form is:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><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><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">The first differences of a quadratic pattern are not constant. However, the differences between consecutive first differences, called the second differences, are constant.<\/p>\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\"><mrow><mn>3<\/mn><mo>;<\/mo><mn>8<\/mn><mo>;<\/mo><mn>15<\/mn><mo>;<\/mo><mn>24<\/mn><mo>;<\/mo><mn>35<\/mn><mo>;<\/mo><mo>\u2026<\/mo><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Calculate the first differences:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mn>8<\/mn><mo>\u2212<\/mo><mn>3<\/mn><mo>=<\/mo><mn>5<\/mn><\/mrow><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mn>15<\/mn><mo>\u2212<\/mo><mn>8<\/mn><mo>=<\/mo><mn>7<\/mn><\/mrow><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mn>24<\/mn><mo>\u2212<\/mo><mn>15<\/mn><mo>=<\/mo><mn>9<\/mn><\/mrow><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mn>35<\/mn><mo>\u2212<\/mo><mn>24<\/mn><mo>=<\/mo><mn>11<\/mn><\/mrow><\/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\"><mrow><mn>5<\/mn><mo>;<\/mo><mn>7<\/mn><mo>;<\/mo><mn>9<\/mn><mo>;<\/mo><mn>11<\/mn><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Calculate the second differences:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mn>7<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mo>=<\/mo><mn>2<\/mn><\/mrow><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mn>9<\/mn><mo>\u2212<\/mo><mn>7<\/mn><mo>=<\/mo><mn>2<\/mn><\/mrow><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mn>11<\/mn><mo>\u2212<\/mo><mn>9<\/mn><mo>=<\/mo><mn>2<\/mn><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">The constant second difference confirms that the pattern is quadratic.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Constant first differences<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Constant first differences indicate a linear or arithmetic pattern. If the common difference is positive, the terms increase. If it is negative, the terms decrease.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For a linear pattern:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msub><mi>T<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mi>p<\/mi><mi>n<\/mi><mo>+<\/mo><mi>q<\/mi><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">The coefficient of the term number equals the constant first difference:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mi>p<\/mi><mo>=<\/mo><mi>d<\/mi><\/mrow><\/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 general term of:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mn>12<\/mn><mo>;<\/mo><mn>17<\/mn><mo>;<\/mo><mn>22<\/mn><mo>;<\/mo><mn>27<\/mn><mo>;<\/mo><mo>\u2026<\/mo><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Calculate the first difference:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mi>d<\/mi><mo>=<\/mo><mn>17<\/mn><mo>\u2212<\/mo><mn>12<\/mn><\/mrow><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mi>d<\/mi><mo>=<\/mo><mn>5<\/mn><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Use the arithmetic sequence formula:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msub><mi>T<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mi>a<\/mi><mo>+<\/mo><mo>(<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>)<\/mo><mi>d<\/mi><\/mrow><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msub><mi>T<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mn>12<\/mn><mo>+<\/mo><mo>(<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>)<\/mo><mo>(<\/mo><mn>5<\/mn><mo>)<\/mo><\/mrow><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msub><mi>T<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mn>12<\/mn><mo>+<\/mo><mn>5<\/mn><mi>n<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msub><mi>T<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mn>5<\/mn><mi>n<\/mi><mo>+<\/mo><mn>7<\/mn><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore, the gradient of the corresponding straight-line graph is five and its y-intercept is seven.<\/p>\n\n\n\n\n<figure class=\"wp-block-image\">\n<a href=\"https:\/\/sqooltutors.co.za\/sign-up-b\/\">\n<img decoding=\"async\" src=\"https:\/\/lmxddlwowqlsyucleuyv.supabase.co\/storage\/v1\/object\/public\/pdf_ads\/batch%201\/Doc%20Image%20Aug%2022,%202026,%2004_22_10%20PM%20(1).png\" alt=\"\" \/>\n<\/a>\n<\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Constant second differences<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Constant second differences indicate a quadratic number pattern. For the general quadratic pattern:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><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><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">The constant second difference is equal to twice the coefficient of the squared term:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mtext>Constant second difference<\/mtext><mo>=<\/mo><mn>2<\/mn><mi>a<\/mi><\/mrow><\/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\"><mrow><mi>a<\/mi><mo>=<\/mo><mfrac><mtext>constant second difference<\/mtext><mn>2<\/mn><\/mfrac><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">The first difference between the first and second terms is:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msub><mi>T<\/mi><mn>2<\/mn><\/msub><mo>\u2212<\/mo><msub><mi>T<\/mi><mn>1<\/mn><\/msub><mo>=<\/mo><mn>3<\/mn><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">The first term is:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msub><mi>T<\/mi><mn>1<\/mn><\/msub><mo>=<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo>+<\/mo><mi>c<\/mi><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">These relationships can be used to calculate the values of the three constants.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Remember<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Constant first differences indicate a linear pattern.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Constant second differences indicate a quadratic pattern.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">If neither the first nor second differences are constant, the pattern may follow another rule.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Finding the general term of a quadratic pattern<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Example<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Find the general term of:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mn>3<\/mn><mo>;<\/mo><mn>8<\/mn><mo>;<\/mo><mn>15<\/mn><mo>;<\/mo><mn>24<\/mn><mo>;<\/mo><mn>35<\/mn><mo>;<\/mo><mo>\u2026<\/mo><\/mrow><\/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\"><mrow><mn>5<\/mn><mo>;<\/mo><mn>7<\/mn><mo>;<\/mo><mn>9<\/mn><mo>;<\/mo><mn>11<\/mn><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">The second differences are:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mn>2<\/mn><mo>;<\/mo><mn>2<\/mn><mo>;<\/mo><mn>2<\/mn><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Start with the general form:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><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><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Use the constant second difference:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mn>2<\/mn><mi>a<\/mi><mo>=<\/mo><mn>2<\/mn><\/mrow><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mi>a<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Use the first first difference:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mn>3<\/mn><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo>=<\/mo><mn>5<\/mn><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Substitute the value of the first coefficient:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mn>3<\/mn><mo>(<\/mo><mn>1<\/mn><mo>)<\/mo><mo>+<\/mo><mi>b<\/mi><mo>=<\/mo><mn>5<\/mn><\/mrow><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mi>b<\/mi><mo>=<\/mo><mn>2<\/mn><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Use the first term:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>3<\/mn><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Substitute the known values:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mn>1<\/mn><mo>+<\/mo><mn>2<\/mn><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>3<\/mn><\/mrow><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore, the general term is:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><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><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Check the rule using the third term:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msub><mi>T<\/mi><mn>3<\/mn><\/msub><mo>=<\/mo><msup><mrow><mo>(<\/mo><mn>3<\/mn><mo>)<\/mo><\/mrow><mn>2<\/mn><\/msup><mo>+<\/mo><mn>2<\/mn><mo>(<\/mo><mn>3<\/mn><mo>)<\/mo><\/mrow><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msub><mi>T<\/mi><mn>3<\/mn><\/msub><mo>=<\/mo><mn>9<\/mn><mo>+<\/mo><mn>6<\/mn><\/mrow><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msub><mi>T<\/mi><mn>3<\/mn><\/msub><mo>=<\/mo><mn>15<\/mn><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">The answer agrees with the original pattern.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Common Mistake<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Do not use the arithmetic sequence formula when only the second differences are constant. The arithmetic formula applies only when the first differences are constant.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Finding specific terms<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Once the general term is known, a specific term can be found by substituting its term number.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Example 1<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Find the twenty-fifth term of the linear pattern with general term:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msub><mi>T<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mn>4<\/mn><mi>n<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Substitute the required term number:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msub><mi>T<\/mi><mn>25<\/mn><\/msub><mo>=<\/mo><mn>4<\/mn><mo>(<\/mo><mn>25<\/mn><mo>)<\/mo><mo>+<\/mo><mn>3<\/mn><\/mrow><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msub><mi>T<\/mi><mn>25<\/mn><\/msub><mo>=<\/mo><mn>100<\/mn><mo>+<\/mo><mn>3<\/mn><\/mrow><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msub><mi>T<\/mi><mn>25<\/mn><\/msub><mo>=<\/mo><mn>103<\/mn><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Example 2<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Find the twentieth term of the quadratic pattern:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><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><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Substitute the required term number:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msub><mi>T<\/mi><mn>20<\/mn><\/msub><mo>=<\/mo><msup><mrow><mo>(<\/mo><mn>20<\/mn><mo>)<\/mo><\/mrow><mn>2<\/mn><\/msup><mo>+<\/mo><mn>2<\/mn><mo>(<\/mo><mn>20<\/mn><mo>)<\/mo><\/mrow><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msub><mi>T<\/mi><mn>20<\/mn><\/msub><mo>=<\/mo><mn>400<\/mn><mo>+<\/mo><mn>40<\/mn><\/mrow><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msub><mi>T<\/mi><mn>20<\/mn><\/msub><mo>=<\/mo><mn>440<\/mn><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Sometimes the term value is given and the term number must be found.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Example 3<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Determine which term of the pattern is equal to one hundred and sixty-eight if:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><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><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Set the general term equal to the given value:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msup><mi>n<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>2<\/mn><mi>n<\/mi><mo>=<\/mo><mn>168<\/mn><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Write the equation in standard form:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msup><mi>n<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>2<\/mn><mi>n<\/mi><mo>\u2212<\/mo><mn>168<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Factorise:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mo>(<\/mo><mi>n<\/mi><mo>+<\/mo><mn>14<\/mn><mo>)<\/mo><mo>(<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mn>12<\/mn><mo>)<\/mo><mo>=<\/mo><mn>0<\/mn><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Solve for the term number:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mi>n<\/mi><mo>+<\/mo><mn>14<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mi>n<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>14<\/mn><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">or<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mi>n<\/mi><mo>\u2212<\/mo><mn>12<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mi>n<\/mi><mo>=<\/mo><mn>12<\/mn><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">A term number must be a positive whole number, so the negative solution is rejected. Therefore, one hundred and sixty-eight is the twelfth term.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Solving problems involving number patterns<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Number-pattern problems may be presented using diagrams, matchsticks, tiles, rows of seats or other practical situations. First identify what changes from one stage to the next. Then write a general term and use it to answer the question.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Example 1: Rows of seats<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The first row of a hall has eighteen seats. Each new row has three more seats than the previous row. Calculate the number of seats in the twentieth row.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The pattern is:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mn>18<\/mn><mo>;<\/mo><mn>21<\/mn><mo>;<\/mo><mn>24<\/mn><mo>;<\/mo><mn>27<\/mn><mo>;<\/mo><mo>\u2026<\/mo><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Identify the first term and common difference:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mi>a<\/mi><mo>=<\/mo><mn>18<\/mn><\/mrow><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mi>d<\/mi><mo>=<\/mo><mn>3<\/mn><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Use the arithmetic sequence formula:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msub><mi>T<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mi>a<\/mi><mo>+<\/mo><mo>(<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>)<\/mo><mi>d<\/mi><\/mrow><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msub><mi>T<\/mi><mn>20<\/mn><\/msub><mo>=<\/mo><mn>18<\/mn><mo>+<\/mo><mo>(<\/mo><mn>20<\/mn><mo>\u2212<\/mo><mn>1<\/mn><mo>)<\/mo><mo>(<\/mo><mn>3<\/mn><mo>)<\/mo><\/mrow><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msub><mi>T<\/mi><mn>20<\/mn><\/msub><mo>=<\/mo><mn>18<\/mn><mo>+<\/mo><mn>19<\/mn><mo>(<\/mo><mn>3<\/mn><mo>)<\/mo><\/mrow><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msub><mi>T<\/mi><mn>20<\/mn><\/msub><mo>=<\/mo><mn>18<\/mn><mo>+<\/mo><mn>57<\/mn><\/mrow><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msub><mi>T<\/mi><mn>20<\/mn><\/msub><mo>=<\/mo><mn>75<\/mn><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore, the twentieth row has seventy-five seats.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Example 2: Matchstick pattern<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Suppose one square requires four matchsticks. When another square is joined in a row, it shares one side with the previous square. Each additional square therefore requires three additional matchsticks.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The pattern is:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mn>4<\/mn><mo>;<\/mo><mn>7<\/mn><mo>;<\/mo><mn>10<\/mn><mo>;<\/mo><mn>13<\/mn><mo>;<\/mo><mo>\u2026<\/mo><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Determine the general term:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mi>a<\/mi><mo>=<\/mo><mn>4<\/mn><\/mrow><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><mi>d<\/mi><mo>=<\/mo><mn>3<\/mn><\/mrow><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msub><mi>T<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mi>a<\/mi><mo>+<\/mo><mo>(<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>)<\/mo><mi>d<\/mi><\/mrow><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msub><mi>T<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mn>4<\/mn><mo>+<\/mo><mo>(<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>)<\/mo><mo>(<\/mo><mn>3<\/mn><mo>)<\/mo><\/mrow><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msub><mi>T<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mn>4<\/mn><mo>+<\/mo><mn>3<\/mn><mi>n<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msub><mi>T<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mn>3<\/mn><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Calculate the number of matchsticks needed for fifty joined squares:<\/p>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msub><mi>T<\/mi><mn>50<\/mn><\/msub><mo>=<\/mo><mn>3<\/mn><mo>(<\/mo><mn>50<\/mn><mo>)<\/mo><mo>+<\/mo><mn>1<\/mn><\/mrow><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\" style=\"margin-top:12px;margin-bottom:12px\"><math display=\"block\"><mrow><msub><mi>T<\/mi><mn>50<\/mn><\/msub><mo>=<\/mo><mn>151<\/mn><\/mrow><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore, one hundred and fifty-one matchsticks are required.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Exam Tip<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Read carefully whether the question asks for a term value, a term number or the general term. Show the differences clearly, state whether the pattern is linear or quadratic, substitute into the correct formula, and reject solutions that are not valid term numbers.<\/p>\n\n\n\n\n<figure class=\"wp-block-image\">\n<a href=\"https:\/\/sqooltutors.co.za\/sign-up-b\/\">\n<img decoding=\"async\" src=\"https:\/\/lmxddlwowqlsyucleuyv.supabase.co\/storage\/v1\/object\/public\/pdf_ads\/batch%201\/Doc%20Image%20Aug%2022,%202026,%2004_21_14%20PM%20(1).png\" alt=\"\" \/>\n<\/a>\n<\/figure>\n","protected":false},"excerpt":{"rendered":"<p>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 called its term number. Number patterns may be represented numerically, algebraically, verbally or graphically. When studying a pattern, identify: The first few terms. The differences between consecutive terms. [&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-256","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"acf":{"document_name":"Number patterns","grade":11,"subject":"Mathematics","term":4,"paper":1},"_links":{"self":[{"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/posts\/256","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=256"}],"version-history":[{"count":0,"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/posts\/256\/revisions"}],"wp:attachment":[{"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/media?parent=256"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/categories?post=256"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/tags?post=256"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}