Find the missing number. 8,12,10,16,12,.....

CONCEPT: An interleaved (alternating) number series is actually two independent arithmetic progressions woven together — the terms sitting at odd positions form one AP with its own common difference, and the terms sitting at even positions form a separate AP with a different common difference. To find a missing term, first identify which position it occupies, then extend the AP that owns that position.

APPLICATION (numbered steps):

1. Number the given terms by position: 1st = 8, 2nd = 12, 3rd = 10, 4th = 16, 5th = 12, 6th = missing term.

2. Odd positions (1st, 3rd, 5th, ...) give 8, 10, 12 — each term is 2 more than the previous one, so this strand continues as 8, 10, 12, 14, ...

3. Even positions (2nd, 4th, 6th, ...) give 12, 16 — each term is 4 more than the previous one, so this strand continues as 12, 16, 20, 24, ...

4. The missing value is the 6th term, which belongs to the even-position strand. Extending that strand one step past 16 gives 16 + 4 = 20.

CROSS-CHECK: Compute the differences between consecutive terms instead: 12-8=+4, 10-12=-2, 16-10=+6, 12-16=-4. These differences follow their own pattern (+4, -2, +6, -4, +8, -6, ...), so the next difference after -4 is +8. Adding it to the 5th term gives 12+8=20, the same result obtained from the two-strand method.

Hence, the missing number is 20.