Question

The ratio of the fifth term from the beginning to the fifth term from the end in the expansion of $\left(\sqrt[4]{2} + \frac{1}{\sqrt[4]{3}}\right)^{n}$ is $\sqrt{6}:1.$ If $n=\frac{20}{\lambda },$ then the value of $\lambda $ is

Answer 1

$\frac{T_{5}}{T_{n - 5 + 2}}= \, \frac{^{n} C_{4} \left(2^{\frac{1}{4}}\right)^{n - 4} \cdot \left(3^{- \frac{1}{4}}\right)^{4}}{^{n} C_{n - 4} \left(2^{\frac{1}{4}}\right)^{4} \cdot \left(3^{- \frac{1}{4}}\right)^{n - 4}}$
$\frac{T_{5}}{T_{n - 5 + 2}}= \, \frac{2^{\frac{n - 4}{4}} \cdot 3^{\frac{n - 4}{4}}}{6}=\sqrt{6}$
$\Rightarrow 6^{\frac{n - 4}{4}}=6^{\frac{3}{2}}$
$\Rightarrow n=10=\frac{20}{\lambda }$
$\Rightarrow \lambda =2$