Question

Find the index n of the binomial ${(\frac{x}{5}+\frac{2}{5})}^n$ if the $9^{th}$ term of the expansion has numerically the greatest coefficient $\left(nϵN\right)$

Answer 1

${(\frac{x}{5}+\frac{2}{5})}^n={\left(\frac{1}{5}\right)}^n{(x+2)}^n$

$\Rightarrow T_{r+1}={\left(\frac{1}{5}\right)}^n{}^{n}C{x}^{n-r}{2}^r$

If the $9th$ term of the expansion has numerically the greatest coefficient,

$\Rightarrow (\frac{n}{2}+1)=9$.

$n=16$ (answer).