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Combination

Let C 0 , C...

Question

Let $C_{0}, C_{1}, . . . . C_{n}$ denotes the binolmial 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

A

2^{n}

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B

2^{- n}

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C

0

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D

1

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Solution

The correct option is C $0$
Expansion of $(1 + x)^{n}$ is $= C_{0} + C_{1} x + C_{2} x^{2} \dots + C_{n} x^{n}$
On differentiating both sides w.r.t $x$ , we get
$n (1 + x)^{x - 1} = C_{1} + 2 C_{2} x + 3 C_{3} x^{2} + \dots + n C_{n} x^{n - 1}$
putting $x = - 1$
$C_{1} - 2 C_{2} + 3 C_{3} - \dots + (- 1)^{n - 1} n C_{n} = 0$

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Similar questions

C_{0}, C_{1}, C_{2}, . . . . . . . C_{n}

denote the coefficients in the expansion of

{(1 + x)}^{n}

C_{1} + 2 C_{2} + 3 C_{3} + . . . . n C_{n} = n . {(2)}^{n - 1}

C_{0}, C_{1}, C_{2}, . . . . . C_{n}

denote the coefficient in the expansion of

(1 + x)^{n}

, then the value of

C_{1} + 2 C_{2} + 3 C_{3} . . . . + n C_{n}

c_{0}, c_{1}, c_{2}, . . . . . c_{n}

denote the coefficients in the expansion of

(1 x)^{n}

c_{1} + 2 c_{2} + 3 c_{3} + . . . . . . + n c_{n} = n {.2}^{n - 1}

c_{0}, c_{1}, c_{2}, . . . . c_{n}

denote the coefficients in the expansion of

(1 + x)^{n}

\frac{c_{1}}{c_{0}} + \frac{2 c_{2}}{c_{1}} + \frac{3 c_{3}}{c_{2}} + . . . . . + \frac{n c_{n}}{c_{n - 1}} = \frac{n (n + 1)}{2}

c_{0}, c_{1}, c_{2}, . . . . . . . c_{n}

denote the coefficients in the expansion of

{(1 + x)}^{n}

\frac{c_{1}}{c_{0}} + \frac{2 c_{2}}{c_{1}} + \frac{3 c_{3}}{c_{2}} + . . . \frac{n c_{n}}{c_{n - 1}} = \frac{n (n + 1)}{2}

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