Find the next number in the following series : 2, 5, 10, 17, 26, 37, 50, ?

Concept

A number series is decoded from its first differences — the gaps between consecutive terms. When those gaps themselves follow a regular progression, the series has a closed form in the position number n.

A run of consecutive odd gaps (3, 5, 7, …) is the signature of a square-based rule, because n2 − (n − 1)2 = 2n − 1.

Application

1. Take the gap between each pair of consecutive given terms: 5 − 2 = 3.

2. Continue: 10 − 5 = 5, then 17 − 10 = 7, then 26 − 17 = 9.

3. Continue: 37 − 26 = 11, then 50 − 37 = 13.

4. The gaps therefore read 3, 5, 7, 9, 11, 13 — consecutive odd numbers rising by 2 each time.

5. The next gap in that run is 13 + 2 = 15.

6. Add it to the last given term: 50 + 15 = 65.

Cross-check

The same series is produced by the closed form n2 + 1, where n is the position of the term. Evaluating it confirms every given term and fixes the eighth one independently of the gap method.

The eighth term is 82 + 1 = 64 + 1 = 65, matching the value obtained by extending the odd-gap run.

Result

The next number in the series is 65.