The missing two numbers in the following series are 1, 4, 3, 16, 5, 36, ?, ?,…

Concept: An interleaved (alternating) number series is really two separate sub-series running through alternate positions. Solve each sub-series on its own — one following an arithmetic rule, the other a squaring rule — then read the missing values back into their original slots.

Application:

1. Split the ten terms by position: the 1st, 3rd, 5th, 7th, 9th terms are 1, 3, 5, ?, 9, and the 2nd, 4th, 6th, 8th, 10th terms are 4, 16, 36, ?, 100.

2. In the first (odd-position) group, each term is 2 more than the one before it: 1 → 3 → 5 → 7 → 9. So the 7th term — the first missing number — is 7.

3. In the second (even-position) group, each term is the square of an increasing even number: 22 = 4, 42 = 16, 62 = 36, 82 = 64, 102 = 100. So the 8th term — the second missing number — is 64.

4. Reading the two missing terms back into the series in order gives 7 and 64.

Cross-check: Test each offered pair against the two sub-series rules independently rather than trusting the pattern by eye. The first number of the missing pair must be the next odd number in 1, 3, 5, …, 9 — that is 7. The second number must be the square of the next even number in 2, 4, 6, …, 10 — that is 82 = 64. Checking each option against both rules: 50 is not the square of any of 2, 4, 6, 8, 10; neither are 72 or 81 (81 is a square, but of the odd number 9, not of an even number in this sequence); and 8 itself is not an odd number, so it cannot be the missing odd-position term at all. Only the pair 7, 64 satisfies both rules simultaneously, confirming the missing numbers are 7 and 64.

Answer: The missing two numbers are 7 and 64.