Question
Let C _ r denote the coefficient of x^ r in the binomial expansion of (1+x)^ n , n N , 0 r n . If P_ n =C_ 0 -C_ 1 + 2^ 2 3 C_ 2 - 2^ 3 4 C_ 3 + . .+ (-2)^ n n+1 C_ n , then the value of _ n=1 ^ 25 1 P_ 2 n equals.
JEE Main
Mathematics
Binomial Theorem
2026
JEE Main 2026 (22 January Shift 2)
JEE Main Mathematics Question (2026) — Solution
Options
- A. 650
- B. 675
- C. 525
- D. 580
Answer
B. 675
Step-by-step solution
Using the identity 1 r+1 n r = 1 n+1 n+1 r+1 , we get P_n = _ r=0 ^ n (-2)^r r+1 C_r = 1 n+1 _ s=1 ^ n+1 (-2)^ s n+1 s . Since (1-2)^ n+1 = (-1)^ n+1 = _ s=0 ^ n+1 (-2)^s n+1 s , we have _ s=1 ^ n+1 (-2)^s n+1 s = (-1)^ n+1 - 1. For even n = 2k: P_ 2k = 1 - (-1) 2(2k+1) = 1 2k+1 . Thus 1 P_ 2n = 2n+1, and _ n=1 ^ 25 (2n+1) = 2 25 26 2 + 25 = 675.
Related: Mathematics — Binomial Theorem · All PYQ Banks