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:
A linear pattern increases or decreases by the same amount between consecutive terms. Therefore, its first differences are constant.
Example
Consider the pattern:
Calculate the first differences:
The constant first difference is:
The pattern is linear because the first differences are constant.
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.
A linear rule has the form:
In this form, the common difference is the gradient and the constant term is the y-intercept of the extended graph.
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.
The common difference is calculated using:
It can also be checked using any two consecutive terms:
Example
Consider the arithmetic sequence:
Calculate the common difference:
This is a decreasing arithmetic sequence because the common difference is negative.
An arithmetic sequence can also be described recursively. A recursive rule states how to obtain each term from the previous term:
For the sequence above:
A recursive rule requires the previous term. A general term allows any term to be calculated directly.
The general term of an arithmetic sequence is:
In this formula, the first term is represented by:
The term number is represented by:
The common difference is represented by:
Example
Find the general term of:
Calculate the common difference:
Identify the first term:
Substitute into the general formula:
Simplify:
Exam Tip
Always test the general term by substituting the first two term numbers.
The calculated values agree with the original sequence.
A quadratic number pattern has a general term containing a squared term. Its general form is:
The first differences of a quadratic pattern are not constant. However, the differences between consecutive first differences, called the second differences, are constant.
Example
Consider the pattern:
Calculate the first differences:
The first differences are:
Calculate the second differences:
The constant second difference confirms that the pattern is quadratic.
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.
For a linear pattern:
The coefficient of the term number equals the constant first difference:
Example
Determine the general term of:
Calculate the first difference:
Use the arithmetic sequence formula:
Therefore, the gradient of the corresponding straight-line graph is five and its y-intercept is seven.
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Constant second differences indicate a quadratic number pattern. For the general quadratic pattern:
The constant second difference is equal to twice the coefficient of the squared term:
Therefore:
The first difference between the first and second terms is:
The first term is:
These relationships can be used to calculate the values of the three constants.
Remember
Constant first differences indicate a linear pattern.
Constant second differences indicate a quadratic pattern.
If neither the first nor second differences are constant, the pattern may follow another rule.
Example
Find the general term of:
The first differences are:
The second differences are:
Start with the general form:
Use the constant second difference:
Use the first first difference:
Substitute the value of the first coefficient:
Use the first term:
Substitute the known values:
Therefore, the general term is:
Check the rule using the third term:
The answer agrees with the original pattern.
Common Mistake
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.
Once the general term is known, a specific term can be found by substituting its term number.
Example 1
Find the twenty-fifth term of the linear pattern with general term:
Substitute the required term number:
Example 2
Find the twentieth term of the quadratic pattern:
Substitute the required term number:
Sometimes the term value is given and the term number must be found.
Example 3
Determine which term of the pattern is equal to one hundred and sixty-eight if:
Set the general term equal to the given value:
Write the equation in standard form:
Factorise:
Solve for the term number:
or
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.
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.
Example 1: Rows of seats
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.
The pattern is:
Identify the first term and common difference:
Use the arithmetic sequence formula:
Therefore, the twentieth row has seventy-five seats.
Example 2: Matchstick pattern
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.
The pattern is:
Determine the general term:
Calculate the number of matchsticks needed for fifty joined squares:
Therefore, one hundred and fifty-one matchsticks are required.
Exam Tip
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.
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