Linear sequence nth terms

For a sequence with first term a and constant difference d, the nth term is a + (n − 1)d. The index of the first listed term is 1. Expanding gives dn + (a − d), which makes it easy to check the first term. Negative differences give decreasing sequences.

To decide whether a value occurs, equate the nth-term expression to that value and solve for n. The answer must be a positive integer. Symbolic first terms work in the same way: subtract consecutive terms to find their common difference, then retain the symbols during simplification. A constant second difference describes a quadratic sequence and needs a different nth-term model.

Worked example

The sequence starts 17, 12, 7, 2, … . Find its nth term and decide whether −98 occurs.

1. The difference is −5, so uₙ = 17 − 5(n − 1) = 22 − 5n.
2. Solve 22 − 5n = −98 to obtain n = 24.
3. The index 24 is a positive integer.

Answer: uₙ = 22 − 5n; −98 is the 24th term