Question

If $(1+ax{)}^n=1+8x+24x^2+...,$ then which of the following is/are correct?

• A. $a=1$
• B. $a=2$
• C. $n=8$
• D. $n=4$

Answer 1

Answer: (B), (D)

$n=4$

Given
$(1+ax{)}^n=1+8x+24x^2+\dots \cdots (i)$
Also,
$(1+ax{)}^n={}^n{C}_0(1{)}^n(ax{)}^0+{}^n{C}_1(1{)}^{n-1}(ax{)}^1+{}^n{C}_2(1{)}^{n-2}(ax{)}^2\cdots \cdots (ii)$

Comparing the coefficients of $x$ and $x^2$, we get
$\Rightarrow {}^n{C}_{1}a=8\text{and}{}^n{C}_2{a}^2=24\Rightarrow na=8\text{and}\frac{n(n-1)}{2}{a}^2=24$
Now,
$\frac{n(n-1)}{2}{a}^2=24\Rightarrow na(na-a)=48\Rightarrow 8(8-a)=48(\because na=8)\Rightarrow a=2\Rightarrow n=4$