Question

If ${17}^{\text{th}}$ and ${18}^{\text{th}}$ terms in the expansion of $(2+a{)}^{50}$ are equal, then the value of $a$ is:

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is B $1$

General term $T_{r+1}={}^{50}{C}_r\times 2^{(50-r)}\times a^r$
Now,
$T_{17}=T_{18}\Rightarrow T_{16+1}=T_{17+1}\Rightarrow {}^{50}{C}_{16}\times 2^{34}\times a^{16}={}^{50}{C}_{17}\times 2^{33}\times a^{17}\Rightarrow a=\frac{2\times {}^{50}{C}_{16}}{{}^{50}{C}_{17}}\Rightarrow a=\frac{2\times 17}{(50-17+1)}=1[\because \frac{{}^n{C}_{r-1}}{{}^n{C}_r}=\frac{r}{n-r+1}]$