Question

Given that n is a positive integer and in the expansion of (1+ax)ⁿ, the coefficient of x is 200. Find the coefficient of x in the expansion of (1+ax)²ⁿ.

Answer 1

$(1+ax)^n:$

$$\dbinom{n}{1}(ax)(1)^{n-1}=nax=200x$$

$$na=200$$

$(1+ax)^{2n}:$

$$\dbinom{2n}{1}(ax)(1)^{2n-1}=2nax=2(200)x=400x$$

The coefficient of $x$ in the expansion of $(1+ax)^{2n}$ is $400.$