Question
The sum of all possible values of n N , so that the coefficients of x, x^ 2 and x^ 3 in the expansion of (1+x^ 2 )^ 2 (1+x)^ n , are in arithmetic progression is :
JEE Main
Mathematics
Binomial Theorem
2026
JEE Main 2026 (23 January Shift 1)
JEE Main Mathematics Question (2026) — Solution
Options
- A. 12
- B. 7
- C. 3
- D. 9
Answer
D. 9
Step-by-step solution
(1+x^2)^2 = 1 + 2x^2 + x^4. Multiply with (1+x)^n: Coefficient of x: n. Coefficient of x^2: n 2 + 2 = n(n-1) 2 + 2. Coefficient of x^3: n 3 + 2n = n(n-1)(n-2) 6 + 2n. For AP: 2 ( n(n-1) 2 +2 ) = n + n(n-1)(n-2) 6 + 2n. Simplifying: n^3 - 9n^2 + 26n - 24 = 0. (n-2)(n-3)(n-4) = 0 n = 2, 3, 4. Sum = 2 + 3 + 4 = 9.
Related: Mathematics — Binomial Theorem · All PYQ Banks