Find the next term of the series: 34, 7, 37, 14, 40, 28, 43, ?

Concept: In an alternating (interleaved) series two independent sequences are woven together — one occupies the odd positions and the other occupies the even positions. Each sub-sequence obeys its own rule, so a term is predicted only from the terms that share its position parity, never from the term standing immediately before it.

Application: separate the two chains by position.

1. Index every term: 34 is 1st, 7 is 2nd, 37 is 3rd, 14 is 4th, 40 is 5th, 28 is 6th, 43 is 7th, and the missing term is 8th.

2. Read the odd-position terms on their own: 34, 37, 40, 43. Their successive differences are 37 − 34 = 3, 40 − 37 = 3 and 43 − 40 = 3, so this chain is arithmetic with common difference 3.

3. Read the even-position terms on their own: 7, 14, 28. Their successive ratios are 14 ÷ 7 = 2 and 28 ÷ 14 = 2, so this chain is geometric with common ratio 2.

4. The blank sits at position 8, which is even, so it extends the geometric chain and not the arithmetic one.

5. Multiply the last even-position term by the common ratio: 28 × 2 = 56.

Cross-check and contrasts:

• Rebuild the full series with 56 in place: 34, 7, 37, 14, 40, 28, 43, 56. The odd-position chain 34, 37, 40, 43 still rises by 3 each time and the even-position chain 7, 14, 28, 56 still doubles each time, so both rules hold from start to finish.

• Reading the even-position chain as a constant +7 progression (7, 14, 21, 28, 35) contradicts the data given: that reading needs 21 at position 6, but the series has 28 there. The even chain doubles, it does not add 7, so 35 is not consistent with the series.

• Adding 3 to 43 gives 46, but 3 is the step of the odd-position chain and 43 is the 7th term; an odd-position rule cannot generate an even-position term.

• Adding 14 to 28 gives 42 and adding 3 to 28 gives 31; neither addition is the doubling that governs the even-position chain.

Answer: 56