Question

Let $C_{0}$, $C_{1},...,$ $C_{n}$ denotes the binomial coefficients in the expansion of $(1 + x)^{n}$. The value of $C_{1} - 2C_{2} + 3C_{3} - 4C_{4} +........$ (upto $n$ terms) is

• A. $2^{n}$
• B. $2^{-n}$
• C. 0
• D. 1

Answer 1

Answer: (C)

Since,
$(1+x)^{n}=C_{0}+C_{1} x+C_{2} x^{2}+\ldots+C_{n} x^{n}$
On differentiating w.r.t. $x$, we get $n(1+x)^{n-1}=C_{1}+2 C_{1} x+\ldots+n \cdot C_{n} x^{n-1}$
Put $x=-1$, we get
$0=C_{1}-2 \cdot C_{1}+\ldots+(-1)^{n-1} \cdot n \cdot C_{n}$