What number would replace the question mark (?) in the series given below? 2,…

Concept

In a number series, inspect the successive differences between adjacent terms before extending the pattern.

When those differences follow their own recurrence, continue that recurrence first and then add the resulting difference to the last term.

Application

1. The successive differences are 5 − 2 = 3, 10 − 5 = 5, 19 − 10 = 9, 36 − 19 = 17, 69 − 36 = 33, and 134 − 69 = 65.

2. These differences follow d(next) = 2d − 1: 5 = 2×3 − 1, 9 = 2×5 − 1, 17 = 2×9 − 1, 33 = 2×17 − 1, and 65 = 2×33 − 1.

3. Therefore the next difference is 2×65 − 1 = 129.

4. Adding it to the last term gives 134 + 129 = 263.

Cross-check

Equivalently, the differences are 2 + 1, 4 + 1, 8 + 1, 16 + 1, 32 + 1, and 64 + 1; the next is 128 + 1 = 129, again producing 263.