Question

The middle term in the expansion of ${\left(\frac{10}{x}+\frac{x}{10}\right)}^{10}$ is

• A. ${}^{10}{C}_5$
• B. ${}^{10}{C}_6$
• C. ${}^{10}{C}_5\frac{1}{x^{10}}$
• D. ${}^{10}{C}_5{x}^{10}$

Answer 1

Answer: (A)

General term $=T_{r+1}={}^{10}{C}_r{\left(\frac{10}{x}\right)}^{10-r}{\left(\frac{x}{10}\right)}^r$
Here $n=10$, which is an even number.
Now, ${\left[\frac{10}{2}+1\right]}^{\text{th }}$ term i.e. $6^{\text{th }}$ term is the middle term.
Hence, middle term $=T_6$
$T_{5+1}={}^{10}{C}_5{\left(\frac{10}{x}\right)}^{10-5}{\left(\frac{x}{10}\right)}^5={}^{10}{C}_5{\left(\frac{10}{x}\right)}^5{\left(\frac{x}{10}\right)}^5={}^{10}{C}_5$