Question

If the third term in the binomial expansion of $(1+x{)}^m$ is $-\frac{1}{8}{x}^2$, then the rational value of m is

• A. 2
• B.
• C. 3
• D. 4

Answer 1

Answer: (B)

$\frac{1}{2}$

Given expression $(1+x{)}^m=1{+}^m{C}_{1}x{+}^m{C}_2{x}^2+......$
Where the third term is ${}^m{C}_2$
So, $m_{c_2}=-\frac{1}{8}$ as given.
$\Rightarrow \frac{m(m-1)}{2}=-\frac{1}{8}$
$\Rightarrow 4m^2-4m+1=0$
$\Rightarrow (2m{)}^2-2(2m\left)\right(1)+(1{)}^2=0$ Using the formula of $(a-b{)}^2=a^2-2ab+b^2$
$\Rightarrow (2m-1{)}^2=0\therefore m=\frac{1}{2}$