Question

Find a if the $17^{th}$ and $18^{th}$ terms of the expansion $(2+a{)}^{50}$ are equal.

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

Answer: (A)

$T_{17}=T_{16+1}={}^{50}{C}_{16}{\left(2\right)}^{50-16}{a}^{16}$
$={}^{50}{C}_{16}{2}^{34}{a}^{16}$
Similarly, $T_{18}=T_{17+1}={}^{50}{C}_{17}{2}^{33}{a}^{17}$
Given that $T_{17}=T_{18}$
So, ${}^{50}{C}_{16}{\left(2\right)}^{34}{a}^{16}$
$={}^{50}{C}_{17}{\left(2\right)}^{33}{a}^{17}$
Therefor $\frac{{}^{50}{C}_{16}\cdot 2^{34}}{{}^{50}{C}_{17}\cdot 2^{33}}=\frac{a^{17}}{a^{16}}$
i.e., $a=\frac{{}^{50}{C}_{16}\times 2}{{}^{50}{C}_{17}}$
$=\frac{50!}{16!\times 34!}\times \frac{17!\times 33!}{50!}\times 2=1$