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 follows the pattern $n^3 - n$ for $n = 2, 3, 4, 5, 6, 7$.

• For $n=2$: $2^3 - 2 = 8 - 2 = 6$ (Note: The first term is 2, which fits $1^3+1$ or simply the start of the series $n^3-n$ starting from $n=2$ gives 6, but let's re-evaluate the pattern based on the given numbers 2, 12, 36, 80, 150).

• Let's check the differences: $12-2=10$, $36-12=24$, $80-36=44$, $150-80=70$.

• The second differences are: $24-10=14$, $44-24=20$, $70-44=26$.

• The third differences are: $20-14=6$, $26-20=6$. Since the third difference is constant, the next second difference is $26+6=32$.

• The next first difference is $70+32=102$.

• Therefore, $X = 150 + 102 = 252$.

Why the other options are wrong

(a) 248: 248 is incorrect because it does not match the calculated next term of 252 derived from the constant third difference pattern.
(c) 258: 258 is incorrect because it exceeds the correct value of 252 by 6, likely resulting from an arithmetic error in summing the differences.
(d) 262: 262 is incorrect because it exceeds the correct value of 252 by 10, which does not align with the established difference 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.