9, 3, 18, 6, 36, 12.............

Concept: When a series alternates between two independent patterns, it is called an interleaved series. Split it by position — the 1st, 3rd, 5th, ... terms form one sub-series, and the 2nd, 4th, 6th, ... terms form another — then find the separate rule governing each sub-series, most often a constant multiplying factor (a geometric progression).

Given: 9, 3, 18, 6, 36, 12, ?

To find: the next (7th) term in the series

1. Mark each term with its position: 9 (1st), 3 (2nd), 18 (3rd), 6 (4th), 36 (5th), 12 (6th).

2. Separate the odd-position terms (1st, 3rd, 5th, 7th, ...) from the even-position terms (2nd, 4th, 6th, ...): odd-position terms are 9, 18, 36, ? and even-position terms are 3, 6, 12.

3. Check the rule within each sub-series: 18 is 9 × 2, and 36 is 18 × 2 — so the odd-position terms form a geometric progression with common ratio 2. Likewise 6 is 3 × 2, and 12 is 6 × 2 — the even-position terms follow the same ratio.

4. Apply the same ratio to extend the odd-position sub-series: the missing 7th term = 36 × 2 = 72.

Cross-check: the same doubling can be verified by addition instead of multiplication — 9 + 9 = 18, 18 + 18 = 36, and 36 + 36 = 72 — which matches the ×2 rule found above and confirms the even-position sub-series (3 + 3 = 6, 6 + 6 = 12) stays consistent throughout.

So, the missing term in the series 9, 3, 18, 6, 36, 12, ..... is 72.