In the number series 1, 2, 6, 15, 31, the next term is _______. – UGC NET Question

The differences between consecutive terms are 1, 4, 9 and 16. These are consecutive perfect squares 1², 2², 3² and 4². To continue the pattern, the next difference should be 5² = 25. Adding 25 to the last term 31 gives 31 + 25 = 56. This preserves the square-based structure of the differences and fits the logic of the series.

Option A:
Option A produces 56, which is exactly 25 more than 31 and makes the difference sequence 1, 4, 9, 16, 25. This is a neat progression of squares, showing a clear higher-level pattern. Hence, 56 is the correct next term in the number series.

Option B:
Option B gives 57, which would correspond to a difference of 26 from 31 and does not match the expected square 25. This slight mismatch is enough to break the precise square-difference rule. Therefore, 57 cannot be accepted as the correct continuation.

Option C:
Option C provides 58, which is 27 more than 31 and again deviates from the exact value of 25 required by the pattern. The differences would no longer correspond to consecutive squares. Thus, 58 is not consistent with the structure.

Option D:
Option D suggests 60, giving a difference of 29 from 31, which is far away from 25 and lacks any special square-based interpretation. Choosing 60 would destroy the elegant second-level pattern. Hence, 60 is not the right answer.