Question

Which term in the expansion of ${[\sqrt[3]{3\left(\frac{a}{\sqrt{b}}\right)}+\sqrt{\left(\frac{b}{\sqrt[3]{3a}}\right)}]}^{21}$ contains $a$ and $b$ to one and same power.

• A. $8^{th}$ term
• B. $9^{th}$ term
• C. ${14}^{th}$ term
• D. ${15}^{th}$ term

Answer 1

The correct option is A $8^{th}$ term

For the above question
$T_{r+1}={}^{21}{C}_r{a}^{7-\frac{r}{3}+\frac{r}{6}}{b}^{\frac{7}{2}-\frac{r}{6}+\frac{r}{2}}$
$={}^{21}{C}_r{a}^{7-\frac{r}{6}}{b}^{\frac{7}{2}+\frac{r}{3}}$
For the powers of a and b to be same
$7-\frac{r}{6}=\frac{7}{2}+\frac{r}{3}$
$\frac{7}{2}=\frac{3r}{6}$
$r=7$
This implies that the term number is 8th.