Question

In the binomial expansion of ${\left(\sqrt[3]{\frac{a}{b}}+\sqrt[3]{\frac{b}{\sqrt{a}}}\right)}^{21},$ the term containing same powers of $a$ and $b$ is

• A. $11^{th}$
• B. $13^{th}$
• C. $12^{th}$
• D. $6^{th}$

Answer 1

Answer: (B)

$t_{r+1}={}^{21}{C}_r{\left(\frac{a}{b}\right)}^{\frac{21-r}{3}}\frac{b^{\frac{r}{3}}}{r}$
$={}^{21}{C}_r{a}^{\frac{21-r}{3}-\frac{r}{6}}{b}^{\frac{r}{3}-\frac{21-r}{3}}$
Since, power of $a$ and $b$ are the same $\Rightarrow \frac{21-r}{3}-\frac{r}{6}=\frac{r}{3}-\frac{21-r}{3}$
$\Rightarrow \frac{2(21-r)}{3}=\frac{r}{3}+\frac{r}{6}=\frac{r}{2}$
$84-4r=3r\Rightarrow 7r=84\Rightarrow r=12$
Hence, the $13^{\text{th }}$ term contains the same powers of $a$ and $b$.