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

Answer: 2

$\frac{T_5}{T_{n-5+2}}=\frac{{}^n{C}_4{\left(2^{\frac{1}{4}}\right)}^{n-4}\cdot {\left(3^{-\frac{1}{4}}\right)}^4}{{}^n{C}_{n-4}{\left(2^{\frac{1}{4}}\right)}^4\cdot {\left(3^{-\frac{1}{4}}\right)}^{n-4}}$
$\frac{T_5}{T_{n-5+2}}=\frac{2^{\frac{n-4}{4}}\cdot 3^{\frac{n-4}{4}}}{6}=\sqrt{6}$
$\Rightarrow 6^{\frac{n-4}{4}}=6^{\frac{3}{2}}$
$\Rightarrow n=10=\frac{20}{\lambda }$
$\Rightarrow \lambda =2$