Q: The value of (n + 2) C0 2n+1 – (n + 1) C12n + n. C2 2n-1 – … is equal to
(A) 4 (1 + n)
(B) 4n
(C) 2n
(D) 2n + 4
Sol. (x – 1)n = C0 xn – C1 xn-1 + Cnxn-2 – ….
x2 (x – 1)n = C0 xn+2 – C1 xn+1 + C2 xn -….
differentiating with respect to x , we get
2x (x – 1)n + x2n (x – 1)n-1 = (n + 2) C0xn+1 – (n + 1) C1 xn + nC2xn-1 – ………
put, x = 2, we get
(n + 2) C0 xn+1 – (n + 1) C12n + n. C22n-1 – …. = 4 + 4n
Hence (A) is the correct answer.
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