Name: Middle Term in Binomial Expansion: Illustration
Uploaded: 2019-07-19
Duration: 3 min 24 s
Description: We have previously studied what Binomial Theorem is. We have learnt its expansion. Let's now take a look at this video to observe the Middle Term in Binomial...

Question

Find the second term after the middle term in the expansion of

{(\frac{x}{2} + \frac{2}{x^{2}})}^{20}

.

Accepted Answer

Given that, the binomial expression is ${(\frac{x}{2} + \frac{2}{x^{2}})}^{20}$ .

On comparing ${(\frac{x}{2} + \frac{2}{x^{2}})}^{20}$ with $(x + a)^{n}$ , we have $x = \frac{x}{2}; a = \frac{2}{x^{2}}; n = 20$ (even).

Thus, the number of terms in the expansion of ${(\frac{x}{2} + \frac{2}{x^{2}})}^{20} = n + 1 = 20 + 1 = 21$ (odd).

So there is only one middle term given by $T_{\frac{n}{2} + 1} = T_{\frac{20}{2} + 1} = T_{11}$ .

We know that, general term in the expansion of $(x + a)^{n}$ be $T_{r + 1} = {}^{n}C_{r} x^{n - r} a^{r}$ .

$∴ T_{r + 1} = {}^{20}C_{r} {(\frac{x}{2})}^{20 - r} {(\frac{2}{x^{2}})}^{r}$ (i)

We want to find second term after the middle term i.e., we want to find $T_{13}$ by putting $r = 12$ in equation (i)

Now, $T_{13} = {}^{20}C_{12} {(\frac{x}{2})}^{8} {(\frac{2}{x^{2}})}^{12} = \frac{20!}{12! 8!} \frac{2^{4}}{x^{16}}$ .

$∴$ The second term after the middle term is $T_{13} = \frac{20!}{12! 8!} \frac{2^{4}}{x^{16}}$ .

Find the second term after the middle term in the expansion of ${(\frac{x}{2} + \frac{2}{x^{2}})}^{20}$ .

Important Questions on Binomial Theorem

Learn and Prepare for any exam you want !

Find the second term after the middle term in the expansion of x2+2x220.

Important Questions on Binomial Theorem

Learn and Prepare for any exam you want !

Find the second term after the middle term in the expansion of ${(\frac{x}{2} + \frac{2}{x^{2}})}^{20}$ .