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 $\frac{^{50}C_{16}\cdot2^{34}}{^{50}C_{17}\cdot2^{33}} = \frac{a^{17}}{a^{16}}$
i.e., $a= \frac{^{50}C_{16}\times2}{^{50}C_{17}}$
$ = \frac{50!}{16! \times 34!}\times\frac{17! \times 33!}{50!}\times2 = 1$