Let C0, C1,..., Cn denotes the binomial coefficients in the expansion of (1 + x)n. The value of C1 - 2C2 + 3C3 - 4C4 +........ (upto n terms) is

Question

Let C0, C1,..., Cn denotes the binomial coefficients in the expansion of (1 + x)n. The value of C1 - 2C2 + 3C3 - 4C4 +........ (upto n terms) is (A) 2n (B) 2-n (C) 0 (D) 1

Tardigrade Admin · Accepted Answer

Since, (1+x)n=C0+C1 x+C2 x2+ ldots+Cn xn On differentiating w.r.t. x, we get n(1+x)n-1=C1+2 C1 x+ ldots+n ⋅ Cn xn-1 Put x=-1, we get 0=C1-2 ⋅ C1+ ldots+(-1)n-1 ⋅ n ⋅ Cn

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

$2^{n}$

$2^{- n}$

0

1

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

Q. Let C0​, C1​,..., Cn​ denotes the binomial coefficients in the expansion of (1+x)n. The value of C1​−2C2​+3C3​−4C4​+........ (upto n terms) is

2n

2−n

0

1

Since, (1+x)n=C0​+C1​x+C2​x2+…+Cn​xn On differentiating w.r.t. x, we get n(1+x)n−1=C1​+2C1​x+…+n⋅Cn​xn−1 Put x=−1, we get 0=C1​−2⋅C1​+…+(−1)n−1⋅n⋅Cn​

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

$2^{n}$

$2^{- n}$

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