Question

The greatest term in the expansion of $(3-5x{)}^{15},whenx=\frac{1}{5}$, is

• A. $6th$
• B. $5th$
• C. $5thand6th$
• D. $4thand5th$

Answer 1

The correct option is B $5th$

$(3-5x{)}^{15}x=1/5$

$\Rightarrow 3^{15}{(1-\frac{5x}{3})}^{15}\Rightarrow 3^{15}{(1-\frac{5}{3}\times \frac{1}{5})}^{15}$

$T_{r+1}$ will be greatest term in this exp

$T_{r+1}>T_r$

${}^{15}{C}_{r+1}{\left(\frac{-1}{3}\right)}^{r+1}{\ge }^{15}{C}_r{\left(\frac{-1}{3}\right)}^r$

$\frac{15!\times (15-r-1)!}{(r+1)!}\left(\frac{-1}{3}\right)\ge \frac{15!(15-r)}{r!}$

$\left(\frac{1}{3}\right)\frac{15-r-1}{r+1}+1\le 0$

$\frac{15-r-1+3r-3}{r+1}\le 0$

$\frac{2r-11}{r+1}\le r\le 11/2\Rightarrow 5^{th}$ term