Question

In the expansion of $(1+ax{)}^n$,$nϵN$,the coefficient of $x$ and $x^2$ are $8$ and $24$ respectively,then

• A. $a=2,n=4$
• B. $a=4,n=2$
• C. $a=2,n=6$
• D. $a=-2,n=4$

Answer 1

The correct option is A $a=2,n=4$

$T_3={}^n{C}_2{a}^2{x}^2=24$
$T_1={}^n{C}_{1}ax=8$
Taking the ratio we get
$\frac{{}^n{C}_{2}ax}{n}=3$
$\frac{(n-1)ax}{2}=3$
Since we are only taking the coefficients, we get
$a(n-1)=6$
$a=\frac{6}{(n-1)}$
Substituting in the equation of $T_1$ ,we get
$n\left(\frac{6}{n-1}\right)=8$
$\Rightarrow 6n=8n-8$
$\Rightarrow 2n=8$
$\Rightarrow n=4$
Hence $a=2$.