Next term in the following sequence is : 2, 12, 36, 80, 150, ?

Correct — option (2), 252.

Method 1 — find a rule in the position n.
n = 1: 1² × 2 = 1 × 2 = 2
n = 2: 2² × 3 = 4 × 3 = 12
n = 3: 3² × 4 = 9 × 4 = 36

n = 4: 4² × 5 = 16 × 5 = 80
n = 5: 5² × 6 = 25 × 6 = 150
n = 6: 6² × 7 = 36 × 7 = 252

The nth term is n² × (n + 1), which is the same as n³ + n².

Method 2 — check with differences.
First differences: 10, 24, 44, 70
Second differences: 14, 20, 26
Third differences: 6, 6 (constant)

Next second difference: 26 + 6 = 32
Next first difference: 70 + 32 = 102
Next term: 150 + 102 = 252

Both methods give 252.

The idea to remember: when the third differences are constant, the terms follow a cubic rule — here n³ + n².

Why the others are wrong

• (1)
  210 — 210 would make the next difference 210 − 150 = 60, smaller than the previous 70. The differences here rise at every step (10, 24, 44, 70), so a drop to 60 breaks the pattern.

  210 is 5 × 6 × 7, a product of three consecutive numbers. That rule, n(n + 1)(n + 2), gives 6 for n = 1, not 2, so it does not fit this series.

• (3)
  258 — 258 is 6 more than the correct term. Test it with differences: First difference: 258 − 150 = 108 Second difference: 108 − 70 = 38 Third difference: 38 − 26 = 12

  The third differences so far are 6 and 6, so 12 breaks the pattern. The rule n² × (n + 1) gives 36 × 7 = 252.

• (4)
  270 — 270 fails the difference check: First difference: 270 − 150 = 120 Second difference: 120 − 70 = 50 Third difference: 50 − 26 = 24

  A constant third difference of 6 is needed. The sixth term by the rule is 6³ + 6² = 216 + 36 = 252, not 270.

A number series can be cracked two ways: find a rule in the position n, or build a difference table.

In a difference table, subtract each term from the next, then repeat on the differences. If the kth differences are constant, the terms fit a polynomial of degree k. Constant first differences mean a linear rule, constant second differences a quadratic, constant third differences a cubic.

For a cubic whose leading term is n³, the constant third difference is 6. That is a quick check that n³ + n² is the right family here.

Key facts

• The nth term of 2, 12, 36, 80, 150 is n² × (n + 1) = n³ + n², so the sixth term is 36 × 7 = 252.
• Cubes 1, 8, 27, 64, 125, 216 plus squares 1, 4, 9, 16, 25, 36 give 2, 12, 36, 80, 150, 252.
• Differences: 10, 24, 44, 70, 102; second differences: 14, 20, 26, 32; third differences: constant 6.
• If the kth differences of a sequence are constant, its terms fit a polynomial of degree k.
• For a cubic with leading term n³, the constant third difference is 6.

Common traps

• A rule must fit every given term. 2 to 12 could be "× 6" or "+ 10", but neither continues to 36.
• One level of differences is not enough here. The first and second differences keep changing; only the third differences are constant.
• A neat-looking product is not a rule. 210 = 5 × 6 × 7, but n(n + 1)(n + 2) gives 6 for n = 1, not 2.

UnlockIAS will link similar questions from RAS Pre 2023 and 2016 here once those papers are published on this site.

• practice — not a real PYQ

  Next term in the following sequence is: 0, 6, 24, 60, 120, ?

  • (a)180
  • (b)196
  • (c)210
  • (d)216
  Answer(3) — The nth term is n³ − n: 1 − 1 = 0, 8 − 2 = 6, 27 − 3 = 24, 64 − 4 = 60, 125 − 5 = 120, so the next is 216 − 6 = 210.

  Check: differences 6, 18, 36, 60 and next 90; 120 + 90 = 210. Option (1), 180, makes the difference 60 again; option (2), 196, is 14² and fits no rule here; option (4), 216, is 6³ without subtracting 6.

• practice — not a real PYQ

  Next term in the following sequence is: 3, 10, 29, 66, 127, ?

  • (a)216
  • (b)218
  • (c)220
  • (d)225
  Answer(2) — The nth term is n³ + 2: 1 + 2 = 3, 8 + 2 = 10, 27 + 2 = 29, 64 + 2 = 66, 125 + 2 = 127, so the next is 216 + 2 = 218.

  Option (1), 216, forgets the + 2; options (3) and (4) break the constant third difference of 6.