Question

If $\left(2+\frac{x}{3}\right)^{55}$ is expanded in the ascending powers of $x$ and the coefficients of powers of $x$ in two consecutive terms of the expansion are equal, then these terms are :

• A. $7^{th}$ and $8^{th}$
• B. $8^{th}$ and $9^{th}$
• C. $28^{th}$ and $29^{th}$
• D. $27^{th}$ and $28^{th}$

Answer 1

Answer: (A)

$\left(2+\frac{x}{3}\right)^{55}$
General term
${ }^{55} C _{ r } \times 2^{55- r } \times\left(\frac{ x }{3}\right)^{ r }$
Let $T _{ r +1}$ and $T _{ r +2}$ are having some co-efficients
$\Rightarrow $ Coff. of $ T_{r+1}=$ Coff. of $ T_{r+2}$
${ }^{55} C_{r} \times 2^{55-r} \times\left(\frac{1}{3}\right)^{r}={ }^{55} C_{r+1} \times(2)^{54- r } \times\left(\frac{1}{3}\right)^{r+1}$
$\Rightarrow r = 6$
$\Rightarrow $ Coff. of $T _{7}=$ Coff. of $T _{8}$