Find the next term of the series. 16, 125, 25, 64, 36, 27, 49, ?

Concept

When a sequence alternates between two rules, separate the terms by position: one subsequence contains the odd-positioned terms and the other contains the even-positioned terms.

Then express each subsequence with a common operation, such as powers whose bases change regularly, before interleaving the two patterns again.

Application

1. Odd positions give 16, 25, 36, 49. These are 42, 52, 62, and 72, so the square bases increase by 1.

2. Even positions give 125, 64, 27, and the missing term. The known values are 53, 43, and 33, so the cube bases decrease by 1.

3. The next even-positioned base is therefore 2, giving 23 = 8.

Cross-check

Interleaving the two subsequences reproduces every given term: 42, 53, 52, 43, 62, 33, 72, 23.

Result

The next term is 8.