How do you find the general term of a sequence?

General formulas versus recursive rules

A sequence is an ordered list of numbers where each entry has a position, labeled n = 1, 2, 3, and so on. To define a sequence, you can write either a recursive rule or a general term formula.

A recursive rule tells you how to get to the next term from the term before it, like a_1 = 4 and a_n = a_(n-1) + 3. It requires starting at the beginning and stepping forward one by one. By contrast, the general term formula—also called the explicit formula—connects the position n directly to the value a_n. For example, a_n = 3n + 1 lets you compute the 100th term immediately: 3(100) + 1 = 301.

Finding the general term step by step

First, list the term positions n = 1, 2, 3, 4 alongside their values. For example, examine the list 5, 8, 11, 14.

Second, find the pattern between positions. If the differences between consecutive terms are constant, multiply n by that difference. Here, the terms increase by 3 each step, so the formula contains 3n. Compare 3n to your actual terms: when n = 1, 3(1) = 3, but the first term is 5. Add 2 to adjust. The formula is a_n = 3n + 2.

Third, test the formula on later terms. Plug in n = 4: 3(4) + 2 = 14, which matches the list.

Checking positions and monotonicity

Once you have a general term, you can test whether a specific value belongs to the sequence. Set a_n equal to the target value and solve for n. If n is a positive integer, the value belongs at that position. If n is a fraction, negative, or not an integer, the number does not belong in the sequence.

You can also determine if a sequence is monotonic. A sequence is increasing if a_(n+1) - a_n > 0 for all n, and decreasing if a_(n+1) - a_n < 0. If the terms alternate signs or rise and fall, the sequence is not monotonic.