Find the missing number in the following series? 3, 5, 5, 19, 7, 41, 9, ?

Concept: When a number series shows an alternating (interleaved) pattern, split it into two sub-series by position — odd-positioned terms (1st, 3rd, 5th, 7th...) and even-positioned terms (2nd, 4th, 6th, 8th...). Each sub-series follows its own independent rule — often a simple arithmetic progression, or a series whose differences themselves grow by a constant amount (a second-order difference pattern). Solve each sub-series separately to find the missing term.

1. Split the given series by position into two interleaved sub-series: odd-positioned terms (1st, 3rd, 5th, 7th) and even-positioned terms (2nd, 4th, 6th, 8th).

2. Odd-positioned sub-series: 3, 5, 7, 9 — each term increases by 2 (a simple arithmetic progression).

3. Even-positioned sub-series: 5, 19, 41, ? — find the first differences: 19 - 5 = 14, and 41 - 19 = 22.

4. Second-order difference (the difference of the differences): 22 - 14 = 8, so the first differences themselves grow by a constant 8 each step.

5. Next first difference = 22 + 8 = 30.

6. Missing term = 41 + 30 = 71.

Cross-check: The odd-positioned sub-series (3, 5, 7, 9) stays a clean +2 arithmetic progression, consistent with the overall structure, and the difference sequence 14, 22, 30 shows a steady +8 growth, confirming the result.