The next term in the series : 2, 7, 28, 63, 126, _______ is

Concept

A power-with-offset series is built by taking a base sequence of perfect powers of the consecutive natural numbers and applying one fixed small correction to each of them. When that correction alternates in sign from one position to the next, the term standing at position n is n3 + 1 when n is odd and n3 − 1 when n is even, that is n3 + (−1)n+1. The general method for any series whose terms grow far faster than their differences suggest is therefore: first identify the base sequence of powers, then read off the small correction, and only then extend the pattern.

Applying it to this series

1. Position 1 is odd, so the correction is +1: 13 = 1 and 1 + 1 = 2, the first given term.

2. Position 2 is even, so the correction is −1: 23 = 8 and 8 − 1 = 7, the second given term.

3. Position 3 is odd: 33 = 27 and 27 + 1 = 28, the third given term.

4. Position 4 is even: 43 = 64 and 64 − 1 = 63, the fourth given term.

5. Position 5 is odd: 53 = 125 and 125 + 1 = 126, the fifth given term.

6. The blank stands at position 6, which is even, so the correction is −1: 63 = 216 and 216 − 1 = 215. The missing term is 215.

Cross-check

A difference table gives no usable rule here: the successive differences are 5, 21, 35 and 63, which follow no simple pattern of their own. That mismatch — terms growing much faster than any tidy difference pattern — is exactly the signal to test a power-based rule instead. The cube rule, by contrast, reproduces every given term:

The cubes bracketing the other offered values are 63 = 216 and 73 = 343: 245, 276 and 296 sit 29, 60 and 80 units above 216 respectively, so none of them lies within one unit of a cube of a natural number. The cubes bracketing 215 itself are 53 = 125 and 63 = 216, and 215 sits exactly one unit below 216 — precisely what the alternating rule requires at the even position 6. The next term is therefore 215.