Series Completion: 8, 9, 8, 7, 10, 9, 6, 11, 10, ?, 12

Concept:

A sequence built by interleaving several simpler series is solved by splitting its terms into groups by position — every k-th term starting from a different offset — and finding the arithmetic pattern within each group separately.

Application:

1. Number the terms 8, 9, 8, 7, 10, 9, 6, 11, 10, ?, 12 as 1st through 11th, and split them into three groups of every 3rd term: Group A = (1st, 4th, 7th, 10th), Group B = (2nd, 5th, 8th, 11th), Group C = (3rd, 6th, 9th).

2. Group A (1st, 4th, 7th, 10th terms): 8, 7, 6, ? — each term is 1 less than the previous one.

3. Group B (2nd, 5th, 8th, 11th terms): 9, 10, 11, 12 — each term is 1 more than the previous one.

4. Group C (3rd, 6th, 9th terms): 8, 9, 10 — each term is also 1 more than the previous one.

5. The missing term is the 10th term, which falls in Group A. Continuing Group A's −1 pattern: 6 − 1 = 5.

Cross-check:

With the missing term as 5, Group A reads 8, 7, 6, 5 (each step −1), while Group B and Group C both independently increase by 1 throughout — all three groups stay internally consistent, confirming the missing term.