2, 3, 3, 5, 10, 13, 39, ?, 172, 177

This is a mixed-operation number series in which addition and multiplication alternate, and the operand used each time an operation type recurs increases by one. To find a missing term, track which operand and operation are due at that position (from the pattern already visible in the given terms) and apply it to the previous term.

1. 2 + 1 = 3

2. 3 × 1 = 3

3. 3 + 2 = 5

4. 5 × 2 = 10

5. 10 + 3 = 13

6. 13 × 3 = 39

7. 39 + 4 = missing term

8. (missing term) × 4 = 172

9. 172 + 5 = 177

Continuing the pattern forward from the recovered term confirms it both ways: multiplying it by four gives 172, and adding five to that gives 177 — exactly the two values already given at the end of the series.

So the missing term is 39 + 4 = 43.