Deduce the position rule of a linear sequence

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Deduce the position rule of a linear sequence

A linear sequence has constant first difference d. Its nth term has form dn + c. Compare the given terms with multiples of d to find the constant adjustment; then verify more than one term.

For 5, 8, 11, 14, the difference is 3. Multiples 3, 6, 9, 12 are two less than the corresponding terms, giving 3n + 2. The rule can test whether a value belongs to the sequence by solving for n and requiring an allowed integer position.

A noninteger or nonpositive position is not automatically a sequence member when indexing starts at 1. The formula describes an algebraic relationship, but the position domain selects the actual terms.

Worked example: Find the nth term of 5, 8, 11, 14, ….

The common difference is 3.

Compare the first term with 3 × 1.

Add the constant difference 2.

Answer: 3n + 2.

Check your understanding: Is 50 a term of this sequence?

Answer: Yes, the 16th.

Why: Solve 3n + 2 = 50 to obtain n = 16.