What is the value of X in the sequence 2, 12, 36, 80, 150, X ?
- (a) 248
- (b) 252 ✓ UPSC's answer
- (c) 258
- (d) 262
Why the answer is (b)
• The sequence 2, 12, 36, 80, 150, X follows the pattern n²(n + 1) for the nth term: 1²×2 = 2, 2²×3 = 12, 3²×4 = 36, 4²×5 = 80, 5²×6 = 150.
• Equivalently, the first differences are 10, 24, 44, 70; the second differences are 14, 20, 26; and the third differences are constant at 6, confirming a cubic (degree-3) pattern.
• Extending the second differences: the next second difference is 26 + 6 = 32, so the next first difference is 70 + 32 = 102.
• Adding this to the last known term gives X = 150 + 102 = 252.
• Cross-checking with the closed form n²(n + 1) at n = 6: 36 × 7 = 252, which matches.
• Therefore the value of X is 252, corresponding to option (b).
Why the other options are wrong
- (a) 248
- 248 does not satisfy the n²(n+1) formula (which gives 252 for n = 6) and does not follow from the constant third-difference of 6 that the sequence exhibits.
- (c) 258
- 258 would require a first difference of 108 from 150, implying a second difference of 38 and a third difference of 12, which breaks the constant third-difference pattern of 6.
- (d) 262
- 262 would require a first difference of 112 from 150, implying a second difference of 42 and a third difference of 16, which is inconsistent with the uniform third difference of 6 observed in the sequence.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2022, held on 5 June 2022. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.