Binomial expansion where $n$ is unknown

Question

In the binomial expansion of $(a+x)^n$, where $n\geq4$, the coefficient of $x^3$ is twice the coefficient of $x^4$.

We are asked to show that $n=2a+3$.

I suppose I should use the binomial theorem, but I'm not sure how here.

Answer 1

Hint:
$$(a+x)^n=\sum_{k=0}^{n}\binom{n}{k}x^ka^{n-k}$$ The coefficient of $x^3$ is $\binom{n}{3}a^{n-3}$, and the coefficient of $x^4$ is $\binom{n}{4}a^{n-4}$. Now just solve the equation $$2\binom{n}{4}a^{n-4}=\binom{n}{3}a^{n-3}$$ in the underterminate $n$