The next term in the series 2, 5, 10, 17, 26, 37, ? is :

Concept — In a number series the first differences between consecutive terms usually reveal the generating rule. When those first differences themselves grow by a constant amount — that is, when the second difference is constant — the terms are produced by a quadratic expression in the position number n, most often of the form n2 + c.

Application

1. Take the first differences of consecutive terms: 5 − 2 = 3, 10 − 5 = 5, 17 − 10 = 7, 26 − 17 = 9 and 37 − 26 = 11.

2. The differences 3, 5, 7, 9, 11 each exceed the previous one by exactly 2, so the second difference is constant and the series is quadratic in n.

3. Continuing that constant gap of 2, the difference that follows 11 must be 11 + 2 = 13.

4. Add this difference to the last stated term: 37 + 13 = 50, which is the seventh term of the series.

Cross-check — Test the quadratic form directly, with n as the position of the term: n = 1 gives 12 + 1 = 2, n = 2 gives 22 + 1 = 5, n = 3 gives 32 + 1 = 10, n = 4 gives 42 + 1 = 17, n = 5 gives 52 + 1 = 26 and n = 6 gives 62 + 1 = 37. Every stated term fits n2 + 1, so the seventh term is 72 + 1 = 49 + 1 = 50, the same value the difference method produced.