Question

Find numerically the greatest term in the expansion of ${(3-5x)}^{11}$ when $x=\frac{1}{5}$

• A. $T_{2+1}$
• B. $T_{3+1}$
• C. Both $A$ & $B$
• D. None of the above

Answer 1

Answer: (C)

Both $A$ & $B$

Since, ${(3-5x)}^{11}=3^{11}{(1-\frac{5x}{3})}^{11}$ Now in the expansion of ${(1-\frac{5x}{3})}^{11},$ we have $\frac{T_{r+1}}{T_r}=\frac{(11-r+1)}{r}|-\frac{5x}{3}|=\frac{(12-r)}{r}|-\frac{5}{3}\times 5|(\because x=\frac{1}{5})$

$=\left(\frac{12-r}{r}\right)\left(\frac{1}{3}\right)=\frac{12-r}{3r}\therefore \frac{T_{r+1}}{T_r}\ge 1\Rightarrow \frac{12-r}{3r}\ge 1\Rightarrow 4r\le 12r\le 3\therefore r=2,3$

$\Rightarrow$ so, the greatest terms are $T_{2+1}$ and $T_{3+1}$