Prove that the co-efficient of $x^{n}$ in the expansion of $(1 + x)^{2 n}$ is twice the co-efficient of $x^{n}$ in the expansion of $(1 + x)^{2 n - 1}$ .

Open in App

Solution

We know that coefficient of $x^{n}$ in the expansion of $(1 + x)^{2 n}$

$= 2 n_{C_{n}} = \frac{(2 n)!}{n! n!} = \frac{2 n (2 n - 1)!}{n (n - 1)! n!}$
$= 2 \frac{(2 n - 1)!}{(n - 1)! n!}$ ...(i)

Also the coefficient of $x^{n}$ is the expansion of $(1 + x)^{2 n - 1}$

$= {2 n - 1}_{C_{n}} = \frac{(2 n - 1)!}{(n - 1)! n!}$ ...(ii)

From (i) and (ii) we see that coefficient of $x^{n}$ in $(1 + x)^{2 n}$ is twice the coefficient of $x^{n}$ in the expansion of $(1 + x)^{2 n - 1}$ .

Suggest Corrections

Similar questions

Q. Prove that the coefficient of

x^{n}

in the expansion of

(1 + x)^{2 n}

is twice the coefficient of

x^{n}

in the expansion of

(1 + x)^{2 n - 1}

Prove that the coefficient of $x^{n}$ in the binomial expansion of $(1 + x)^{2 n}$ is twice the coefficient of $x^{n}$ in the binomial expansion of $(1 + x)^{2 n - 1}$ .

Q. The Greatest co-efficient in the expansion of

(1 + x)^{2 n + 2}

Q. If

| x | < 1

then the co-efficient of

x^{n}

in the expansion of

(1 + x + x^{2} + . . . . .)^{2}

Q. The sum of the co-efficients in the expansion of

{(1 - 2 x + 5 x^{2})}^{n}

a

and the sum of the co-efficients in the expansion of

{(1 + x)}^{2 n}

is b then

Join BYJU'S Learning Program

Prove that the co-efficient of xn in the expansion of (1+x)2n is twice the co-efficient of xn in the expansion of (1+x)2n−1.

Prove that the co-efficient of $x^{n}$ in the expansion of $(1 + x)^{2 n}$ is twice the co-efficient of $x^{n}$ in the expansion of $(1 + x)^{2 n - 1}$ .