Mathematics - Binomial Theorem Question with Solution | TestHub

In expansion of $(a+b{)}^n$ the coefficient of $2^{nd},3^{\text{rd }}$ and $4^{\text{th }}$ term are in A.P. which gives $n=7$ also ${}^7{C}_2{a}^5{b}^2=84\Rightarrow a^5{b}^2=4$

Now $\frac{{\log }_2{a}^5+{\log }_2{b}^2}{2}\ge {\left({\log }_2{a}^5\cdot {\log }_2{b}^2\right)}^{1/2}\Rightarrow k\le {\left(\frac{{\log }_2{a}^5{b}^2}{2}\right)}^2$

$k\le 1\Rightarrow$ maximum value of $k$ is 1.