Question

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

• A. $355\times 3^{12}$
• B. $455\times 3^{11}$
• C. $455\times 3^{12}$
• D. $355\times 3^{11}$

Answer 1

The correct option is C $455\times 3^{12}$

$(3-5x{)}^{15}=(3-1{)}^{15}(\because x=\frac{1}{5})$
Now for greatest term
$\left|\begin{array}{c}\frac{T_{r+1}}{T_r}\end{array}\right|=\frac{15-r+1}{r}\left|\begin{array}{c}-\frac{1}{3}\end{array}\right|\ge 1$
$\Rightarrow 16-r\ge 3r$
$\Rightarrow r\le 4$
Hence, $T_4$ and $T_5$ are the numerically greatest terms.
$T_4={}^{15}{C}_3{3}^{(15-3)}(-1{)}^3=-455\times 3^{12}$
and $T_5={}^{15}{C}_4{3}^{(15-4)}(-1{)}^4=455\times 3^{12}$
Also, $|T_4|=|T_5|=455\times 3^{12}$.