Which number will complete the series? 0, 1, 4, 15, ?, 325, 1956

Concept

Many number series are built recursively, where each term is generated from the one before it by combining multiplication and addition, and the multiplier/addend itself increases by 1 at every step. Such a series follows the general rule: next term = (previous term × k) + k, where k = 1, 2, 3, 4, ... for the 1st, 2nd, 3rd, 4th, ... step respectively.

Application

1. Start with the first term: 0.

2. Step 1 (k = 1): 0 × 1 + 1 = 1.

3. Step 2 (k = 2): 1 × 2 + 2 = 4.

4. Step 3 (k = 3): 4 × 3 + 3 = 15.

5. Step 4 (k = 4): 15 × 4 + 4 = 64 — this is the missing term.

6. Step 5 (k = 5): 64 × 5 + 5 = 325.

7. Step 6 (k = 6): 325 × 6 + 6 = 1956.

Cross-check

Continuing the same rule from 64 exactly reproduces the two subsequent given terms — 64 × 5 + 5 = 325, and 325 × 6 + 6 = 1956 — matching the series precisely, which confirms 64 is the correct term.

Therefore, the missing number is 64.