Find the number that can replace the question mark (?) in the series: 4, 27,…

Concept

A number series may interleave two rules. Separate alternate positions and express each term as a power of its base; then identify the rule governing each subsequence.

Application

1. The first, third, fifth, and seventh terms are 4 = 2 × 2, 16 = 4 × 4, 36 = 6 × 6, and 64 = 8 × 8. These are squares of consecutive even numbers.

2. The second, fourth, sixth, and eighth terms are 27 = 3 × 3 × 3, 125 = 5 × 5 × 5, ?, and 729 = 9 × 9 × 9. These are cubes of consecutive odd numbers.

3. Therefore the missing odd base is 7, and 7 × 7 × 7 = 343.

Cross-check

Substituting 343 gives the odd-position bases 2, 4, 6, 8 with square terms and the even-position bases 3, 5, 7, 9 with cube terms, so both interleaved subsequences are consistent.

The missing number is 343.