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 $T_8$