Complete the pattern: 2, 12, 36, 80, 150, ?

Correct — C.
Rule: term n = n² × (n + 1), the same as n³ + n².

1² × 2 = 2, 2² × 3 = 12, 3² × 4 = 36
4² × 5 = 80, 5² × 6 = 150
Next: 6² × 7 = 36 × 7 = 252

Check by differences: 10, 24, 44, 70 grow by 14, 20, 26, so the next grows by 32. 70 + 32 = 102, and 150 + 102 = 252 → option (c).

Why the others are wrong

• (a)
  246 — 246 repeats the last second difference, 26, instead of raising it to 32: 70 + 26 = 96 and 150 + 96 = 246. The rule gives 6² × 7 = 252.
• (b)
  240 — 240 needs a jump of 90 from 150. The differences 10, 24, 44, 70 call for 102 next, and 6² × 7 confirms 252.
• (d)
  258 — 258 needs a jump of 108 from 150, six more than the 102 the differences call for. 6² × 7 = 252 settles it.

When a series grows faster and faster with no constant difference or ratio, test powers of the position number. Here each term is n² × (n + 1): 150 is 5² × 6, so the 6th term is 6² × 7.

The method of differences checks the answer without the formula. First differences 10, 24, 44, 70 have second differences 14, 20, 26, which rise by a steady 6.

Key facts

• The terms are n³ + n²: 1 + 1, 8 + 4, 27 + 9, 64 + 16, 125 + 25.
• For n = 6: 216 + 36 = 252.
• Second differences 14, 20, 26 rise by 6, so the next is 32.

Common traps

• Carrying the second difference 26 forward unchanged, which gives 246
• Testing squares alone or cubes alone, when each term needs both

No directly related past PYQ was found.