Question

The first 3 terms in the expansion of $(1 + ax)^n$ $(n \ne 0)$ are $1$, $6x$ and $16x^2$. Then the values of $a$ and $n$ respectively are

• A. $2,\,9$
• B. $3,\,2$
• C. $2/3,\,9$
• D. $3/2,\,6$

Answer 1

Answer: (C)

In binomial expansion of $(1 + ax)^n$
$T_{2} = \,{}^{n}C_{1}\left(ax\right)^{1}$
$\Rightarrow n\, a\, x = 6x$
$\Rightarrow na = 6\quad\ldots\left(1\right)$
$\Rightarrow n^{2} \,a^{2} = 36$ (after squaring) $\quad\ldots\left(2\right)$
Also, $\frac{n\left(n-1\right)}{2}a^{2}\,x^{2} = 16x^{2}$
$\Rightarrow n\left(n - 1\right) a^{2} = 32\quad\ldots\left(3\right)$
Dividing $\left(3\right)$ by $\left(2\right)$ gives
$\frac{n\left(n-1\right)}{n^{2}} = \frac{32}{36}$
$\Rightarrow \frac{n-1}{n} = \frac{8}{9}$
$\Rightarrow n=9$
$\therefore $ From $\left(1\right)$, we have $9a = 6$
$\Rightarrow a = \frac{2}{3}$