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

By using binomial theorem, expand the following:

${(\frac{x}{3} + \frac{1}{x})}^{5}$

Accepted Answer

Given, ${(\frac{x}{3} + \frac{1}{x})}^{5}$

We know that, $(a + b)^{n} = {}^{n}C_{0} a^{n} + {}^{n}C_{1} a^{n - 1} b + {}^{n}C_{2} a^{n - 2} b^{2} + \dots + {}^{n}C_{r} a^{n - r} b^{r} + \dots \cdot \dots + {}^{n}C_{n} b^{n}$

$∴ {(\frac{x}{3} + \frac{1}{x})}^{5} = {}^{5}C_{0} {(\frac{x}{3})}^{5} + {}^{5}C_{1} {(\frac{x}{3})}^{4} (\frac{1}{x}) + {}^{5}C_{2} {(\frac{x}{3})}^{3} {(\frac{1}{x})}^{2} + {}^{5}C_{3} {(\frac{x}{3})}^{2} {(\frac{1}{x})}^{3} + {}^{5}C_{4} (\frac{x}{3}) {(\frac{1}{x})}^{4} + {}^{5}C_{5} {(\frac{1}{x})}^{5}$

$= 1 (\frac{x^{5}}{24^{3}}) + 5 (\frac{x^{3}}{81}) + 10 (\frac{1}{27}) + 10 (\frac{1}{9 x}) + 5 (\frac{1}{3 x^{3}}) + 1 (\frac{1}{x^{5}})$ $[∵ {}^{n}C_{r} = \frac{n!}{r! (n - r)!} = \frac{n (n - 1) (n - 2) - (n - r + 1)}{r!}]$

$∴ {(\frac{x}{3} + \frac{1}{x})}^{5} = \frac{x^{5}}{243} + \frac{5 x^{3}}{81} + \frac{10}{27} + \frac{10}{9 x} + \frac{5}{3 x^{3}} + \frac{1}{x^{5}}$ .

By using binomial theorem, expand the following:

${(\frac{x}{3} + \frac{1}{x})}^{5}$

Important Questions on Binomial Theorem

Learn and Prepare for any exam you want !

By using binomial theorem, expand the following: x3+1x5

Important Questions on Binomial Theorem

Learn and Prepare for any exam you want !

By using binomial theorem, expand the following:

${(\frac{x}{3} + \frac{1}{x})}^{5}$