What should replace the question mark in the series: 2, 4, 12, 44, ?, 684?

Concept

In a difference-based number series, first subtract consecutive terms and look for a rule among the differences.

If the differences form a geometric sequence, each next difference is obtained by multiplying the previous difference by a fixed ratio.

Application

1. The first difference is 4 − 2 = 2.

2. The second difference is 12 − 4 = 8 = 2 × 4.

3. The third difference is 44 − 12 = 32 = 8 × 4.

4. Therefore, the next difference is 32 × 4 = 128, so the missing term is 44 + 128 = 172.

5. The following difference should be 128 × 4 = 512, and 172 + 512 = 684.

Cross-check

The complete difference sequence is 2, 8, 32, 128, 512; every term is four times the preceding one. Hence the missing value is 172.