Question

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

• A. 2
• B. 0
• C. 1
• D. 3

Answer 1

Answer: (C)

1

In ${(x+a)}^n$, we have $T_{r+1}{=}^n{C}_r{x}^{n-r}{a}^r$ .....(i)

Given, ${17}^{th}$ and ${18}^{th}$ terms are equal.

Thus $T_{17}=T_{18}$

$\Rightarrow T_{16+1}=T_{17+1}$

${\Rightarrow }^{50}{C}_{16}{\left(2\right)}^{50-16}{a}^{16}{=}^{50}{C}_{17}{\left(2\right)}^{50-17}{a}^{17}$ ,,,,,[From (i)]

$\Rightarrow \frac{50!}{16!(50-16)!}{2}^{44}{a}^{16}=\frac{50!}{17!(50-17)!}{2}^{43}{a}^{17}$

$\Rightarrow a=1$