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

Open in App

Solution

We know that
General term of $(a + b)^{n}$ is
$T_{r + 1} =^{n} C_{r} a^{n - r}, b^{r}$
For $(1 + x)^{2 n}$
General term of $(1 + x)^{2 n}$
Putting $a = 1, b = x, n = 2 n$
$T_{r + 1} =^{2 n} C_{r} 1^{2 n - r} (x)^{r}$
$T_{r + 1} =^{2 n} C_{r} x^{r}$ ……………. $(1)$
For coefficient of $x^{n}$
Putting $r = n$ in $(1)$
$T_{r + 1} =^{2 n} C_{n} x^{n}$
Coefficient of $x^{n} =^{2 n} C_{n}$
For $(1 + x)^{2 n - 1}$
General term of $(1 + x)^{2 n - 1}$
Putting $a = 1, b = x, n = 2 n - 1$
$T_{r + 1}^{2 n - 1} C_{n} x^{n}$
Coefficient of $x^{n} =^{2 n - 1} C_{n}$
We have to prove
Coefficient of $x^{n}$ in $(1 + x)^{2 n} = 2 x$ , coefficient of $x^{n}$ in $(1 + x)^{2 n - 1}$
i.e., $^{2 n} C_{n} =^{2 n - 1} C_{n}$
L.H.S R.H.S
$^{2 n} C_{n}$ $2 \times^{2 n - 1} C_{n}$
$= \frac{2 n!}{n! (2 n - n)!}$ $= 2 \times \frac{(2 n - 1)!}{n! (2 n - 1 - n)!}$
$= \frac{2 n!}{n! \cdot n!}$ $= 2 \times \frac{(2 n - 1)!}{n! (n - 1)!}$
Multiply & divide by n
$= 2 \times \frac{(2 n - 1)!}{n! (n - 1)!} \times \frac{n}{n}$
$= \frac{2 n (2 n - 1)!}{n! n (n - 1)!}$
$= \frac{2 n!}{n! n!}$
Hence L.H.S $=$ R.H.S
We proved.

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 coefficient

x^{n}

in the expression of

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

and

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

are in the ratio.

Q. If the coefficient of

x^{n}

(1 + x)^{2 n}

is '

a

' and the coefficient of

x^{n}

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

b

, then

\frac{a}{b} =

Q. 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}

Join BYJU'S Learning Program

Prove that the coefficient of xn in the expression of (1+x)2n is twice the coefficient of xn in the expression of (1+x)2n−1.

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