Name: Binomial Expansion: Illustration
Uploaded: 2019-07-19
Duration: 4 min 17 s
Description: We already know what Binomial Theorem is. Let's watch this video to observe an illustration that will help us understand Binomial Expansion better.

Question

Show that the middle term in the expansion of

(1 + x)^{2 n}

is

\frac{1.3.5 \dots (2 n - 1)}{n!} {(2 x)}^{n}

.

Accepted Answer

Given, number of terms $= 2 n$ , which is even

So, middle term $= {(\frac{2 n}{2} + 1)}^{t h}$ term $= {(n + 1)}^{t h}$ term

Hence, we need to find $T_{n + 1}$

We know that, general term of ${(a + b)}^{n}$ is $T_{r + 1} = {}^{n}C_{r} a^{n - r} b^{r}$

$T_{n + 1} = {}^{2 n}C_{n} {(1)}^{2 n - n} {(x)}^{n}$

$= \frac{(2 n)!}{n! (2 n - n)!} \cdot {(1)}^{n} \cdot x^{n}$

$= \frac{(2 n)!}{n! n!} \cdot x^{n}$

$= \frac{2 n (2 n - 1) (2 n - 2) \dots \dots . 4 \times 3 \times 2 \times 1}{n! n!} x^{n}$

$= \frac{2 n (2 n - 1) (2 n - 2) \dots \dots . 4 \times 3 \times 2 \times 1}{n! n!} x^{n}$

$= \frac{[1 \times 3 \times 5 \times \dots \dots \times (2 n - 3) (2 n - 1)] [2 \times 4 \times 6 \dots . \times (2 n - 2) \times 2 n]}{n! n!} x^{n}$

$= \frac{[1 \times 3 \times 5 \dots \dots \times (2 n - 3) (2 n - 1)] [(2 \times 1) \times (2 \times 2) \times (2 \times 3) \times \dots \times 2 (n - 1) \times 2 (n)]}{n! n!} x^{n}$

$= \frac{[1 \times 3 \times 5 \dots \dots \times (2 n - 3) (2 n - 1)] (2 \times 2 \times 2 \times 2 \dots \dots \times 2) [1 \times 2 \times 3 \dots (n - 1) n]}{n! n!} x^{n}$

$= \frac{[1 \times 3 \times 5 \dots \dots \times (2 n - 3) (2 n - 1)] 2^{n} [1 \times 2 \times 3 \dots \dots (n - 1) n]}{n! n!} x^{n}$

$= \frac{[1 \times 3 \times 5 \dots \dots \times (2 n - 3) (2 n - 1)] 2^{n} (n!)}{n! n!} \cdot x^{n}$

$= \frac{1 \times 3 \times 5 \dots \dots \times (2 n - 1)}{n!} 2^{n} \cdot x^{n}$

Hence, middle term of expansion ${(1 + x)}^{n}$ is $\frac{1 \cdot 3 \cdot 5 \dots \dots \dots (2 n - 1)}{n!} {(2 x)}^{n}$ .

Hence, proved.

Show that the middle term in the expansion of $(1 + x)^{2 n}$ is $\frac{1.3.5 \dots (2 n - 1)}{n!} {(2 x)}^{n}$ .

Important Questions on Binomial Theorem

Learn and Prepare for any exam you want !

Show that the middle term in the expansion of (1+x)2n is 1.3.5…(2n-1)n!2xn.

Important Questions on Binomial Theorem

Learn and Prepare for any exam you want !

Show that the middle term in the expansion of $(1 + x)^{2 n}$ is $\frac{1.3.5 \dots (2 n - 1)}{n!} {(2 x)}^{n}$ .