Find the next number : 2,12,36,80,150,__

CONCEPT: When a number series does not follow a simple arithmetic or geometric rule, check whether each term equals a fixed algebraic expression in its position n (such as n2 or n3, or a sum like n2 + n3). Once that position-to-value rule fits every given term, the missing term is obtained by evaluating the same expression at the required position, and the rule can be verified independently using the method of successive differences, where the order at which differences become constant reveals the degree of the underlying polynomial.

1. Assign position index n = 1, 2, 3, 4, 5 to the five given terms 2, 12, 36, 80, 150, and n = 6 to the term being found.

2. At n = 1: 12 + 13 = 1 + 1 = 2, matching the first term.

3. At n = 2: 22 + 23 = 4 + 8 = 12, matching the second term.

4. At n = 3: 32 + 33 = 9 + 27 = 36, matching the third term.

5. At n = 4: 42 + 43 = 16 + 64 = 80, matching the fourth term.

6. At n = 5: 52 + 53 = 25 + 125 = 150, matching the fifth term.

7. With the rule confirmed across all five given terms, apply it at n = 6: 62 + 63 = 36 + 216 = 252.

Cross-check with successive differences: the gaps between consecutive terms are 10, 24, 44, 70; the gaps between those gaps are 14, 20, 26; and the gaps between those are 6 and 6 — a constant third-level difference, confirming a degree-3 (cubic) pattern, consistent with the n3 term above. Extending the pattern: the next third-level gap stays 6, so the next second-level gap is 26 + 6 = 32, the next first-level gap is 70 + 32 = 102, and the next term is 150 + 102 = 252 — matching the algebraic result exactly.

So the next number in the series is 252.