What is the next number in the series? 1, 10, 27, 64, 66, 145, ?

Concept: When a numeric series does not follow a single fixed rule from one term to the next, check whether it is an interleaved (alternate-term) series — split it into two separate sub-series using the odd-positioned terms and the even-positioned terms, and find the pattern governing each sub-series independently.

Application: Splitting the given series 1, 10, 27, 64, 66, 145, ? by position:

1. Odd-positioned terms (1st, 3rd, 5th, 7th ...): 1, 27, 66, ?

2. Even-positioned terms (2nd, 4th, 6th ...): 10, 64, 145

3. In the odd-positioned sub-series: 1 + 13×2 = 27

4. 27 + 13×3 = 66

5. 66 + 13×4 = 118 — this is the required 7th term

6. In the even-positioned sub-series (confirming the pattern style): 10 + 27×2 = 64

7. 64 + 27×3 = 145

Cross-check: Both sub-series follow the same structural rule — add (base value) × (an increasing multiplier starting at 2). Since the missing 7th term belongs to the odd-positioned sub-series, continuing its established pattern gives 66 + 13×4 = 118, which does not match any of the other offered values under either sub-series' rule.

Hence, the next number in the series is 118.