Question

The first three terms in the expansion of $(1+a{)}^n$ are $t_1=1,t_2=-18,t_3=144$.

Use the general term to determine a and n.

• A. $a=-3,n=9$
• B. $a=-2,n=9$
• C. $a=-3,n=8$
• D. $a=-2,n=8$

Answer 1

The correct option is B $a=-2,n=9$

As we can see from option that n is positive so we proceed with the expansion

$T_1$ $=$ ${}^n{C}_0$ $=1$

$T_2$ $=$ ${}^n{C}_{1}a$ $=n.a=-18$ ....(1)

$T_3$ $=$ ${}^n{C}_2{a}^2$ $=$ $\frac{\left(n\right)(n-1)}{2}$.$a^2$ $=144$ ....(2)

$\Rightarrow$$\left(na\right)(na-a)$.$=288$

Putting the value of $na$ from (1), we get

$(-18)(-18-a)$ $=288$

$\Rightarrow$$-18-a$ $=-16$

$\Rightarrow$ $-a=2$

So, $a=-2$

Substitute this value of $a$ in equation (1), we get

$na=-18$

$\Rightarrow n(-2)=-18$

$\Rightarrow n=9$

Thus value of $a$ and $n$ are $-2$ and $9$.