7.1Patterns in Number Sequences

A number sequence is an ordered list of numbers that follows a specific rule. Each number in the sequence is called a term, labelled $T_1, T_2, T_3, \ldots$ where $T_1$ is the 1st term.

An arithmetic sequence has a constant difference between consecutive terms, called the common difference $d$ . Example: $4, 9, 14, 19, \ldots$ has $d = 5$ .
A geometric sequence has a constant ratio between consecutive terms, called the common ratio $r$ . Example: $3, 6, 12, 24, \ldots$ has $r = 2$ .
Other sequences follow different rules, look for differences of differences, alternating signs, or each term depending on two previous terms.

Finding the rule: arithmetic (increasing)

State the rule and find the next two terms: $\displaystyle 6, 13, 20, 27, 34, \ldots$

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Common difference

= 13 - 6 = 7

Check:

20 - 13 = 7

27 - 20 = 7

✓

Rule: Start with 6, add 7 to get each successive term.

Next terms:

34 + 7 = 41

41 + 7 = 48

Finding the rule: arithmetic (decreasing)

State the rule and find the next two terms: $\displaystyle 55, 48, 41, 34, 27, \ldots$

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Common difference

= 48 - 55 = -7

Rule: Start with 55, subtract 7 to get each successive term.

Next terms:

27 - 7 = 20

20 - 7 = 13

Finding the rule: geometric sequence

State the rule and find the next two terms: $\displaystyle 3, 12, 48, 192, 768, \ldots$

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Common ratio

= \frac{12}{3} = 4

Check:

\frac{48}{12} = 4

\frac{192}{48} = 4

✓

Rule: Start with 3, multiply each term by 4.

Next terms:

768 \times 4 = 3072

3072 \times 4 = 12288

Now you try

Find the next term: $42, 39, 36, 33, 30, \ldots$

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42, 39, 36, 33, 30, \ldots

\displaystyle \text{Common difference}

\displaystyle = 39 - 42 = -3

\displaystyle \text{Next term}

\displaystyle = 30 - 3

\displaystyle = 27

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To name a sequence, test in this order:

Test	If it holds
subtract each term from the next	equal differences means arithmetic
divide each term by the one before	equal ratios means geometric
neither	try differences of the differences, or an alternating rule