Question

For which of the following values of $x$, $7^{\text{th}}$ term is the numerically greatest term in the expansion of ${(1+\frac{2x}{5})}^{12}$

• A. $\frac{9}{4}$
• B. $\frac{5}{2}$
• C. $\frac{11}{4}$
• D. $\frac{13}{4}$

Answer 1

Answer: (A), (B), (C)

The correct options are
A $\frac{9}{4}$
B $\frac{5}{2}$
C $\frac{11}{4}$

$T_7$ is the greatest term in the expansion of ${(1+\frac{2x}{5})}^{12}$
Then,
$\left|\frac{T_7}{T_6}\right|>1$ and $\left|\frac{T_8}{T_7}\right|<1$

Now,
$\frac{T_{r+1}}{T_r}=\frac{n-r+1}{r}\times \frac{2x}{5}$
$\left|\frac{T_7}{T_6}\right|=\frac{12-6+1}{6}\times \frac{2x}{5}$ ($r=6)$
$\frac{7x}{15}>1$
$x>\frac{15}{7}$

Also,
$\left|\frac{T_8}{T_7}\right|=\frac{12-7+1}{7}\times \frac{2x}{5}$
$\frac{12x}{35}<1$
$x<\frac{35}{12}$
$x\in (\frac{15}{7},\frac{35}{12})$