Question

The largest term in the expansion of $(3+2 x )^{50}$, where $x =\frac{1}{5}$, is
$I$. $5^{\text {th }}$
$II$. $3^{\text {rd }}$
$III$. $7^{\text {th }}$
$IV$. $6^{\text {th }}$
Choose the correct option

• A. Only I
• B. Only II
• C. Both I and IV
• D. Both III and IV

Answer 1

Answer: (D)

$\because(3+2 x)^{50}=3^{50}\left(1+\frac{2 x}{3}\right)^{30}$
Here, $T _{ r +1}=3^{50}\,{}^{50} C _{ r }\left(\frac{2 x }{3}\right)^{ r -1}$
and $T_{r+1} = 3^{50} \,{}^{50}C_{r} \left(\frac{2x}{3}\right)^{r-1}$
But $x=\frac{1}{5}\,\,\,\,\,\,\,\,$[given]
$\therefore \frac{ T _{ r +1}}{ T _{ r }} \geq 1 $
$\Rightarrow \frac{{ }^{50} C _{ r }}{{ }^{50} C _{ r -1}} \frac{2}{3} \cdot \frac{1}{5} \geq 1$
$\Rightarrow 102-2 r \geq 15 r $
$\Rightarrow r \leq 6 $
$\Rightarrow r=6$
Therefore, there are two greatest terms $T _{ r }$ and $T _{ r +1}$ i.e., $T _{6}$ and $T _{7}$