1.4: Coefficients in a binomial expansion

(a) Expand $\displaystyle (1+ax)^n$ in ascending powers of $\displaystyle x$ up to and including the term in $\displaystyle x^3$. Express each coefficient of $\displaystyle x$ in terms of $\displaystyle a$ and $\displaystyle n$, where $\displaystyle a$ and $\displaystyle n$ are constants and $\displaystyle n>2$.

The coefficient of $\displaystyle x$ is $\displaystyle 15$ and the coefficient of $\displaystyle x^2$ is equal to the coefficient of $\displaystyle x^3$.

(b) Find the value of $\displaystyle a$ and the value of $\displaystyle n$.

(c) Find the coefficient of $\displaystyle x^3$.