Question

Find the value of $a_1$ for which the coefficient of the middle term in the expansions of $(1+ax{)}^4$and $(1-ax{)}^6$ are equal.

Answer 1

In the expansion $(1+ax{)}^4,$

middle term =${}^4{C}_2(ax{)}^2=6a^2{x}^2$

In the expansion of $(1-ax{)}^6,$

Middle term=${}^6{C}_3(-ax{)}^3=20a^3\times x^3$

It is given that,

Coefficients of middle term $1+ax{)}^4$

=Coefficients of middle term$1-ax{)}^6$

$\Rightarrow 6a^2=-20a^3$

$\Rightarrow a=\frac{6}{-20}=-\frac{3}{10}$

Hence, the value of a is $-\frac{3}{10}$.