A level maths binomial expansion question HELP

Hi, so I recently came across this question regarding Binomial expansion (attached below) and I was wondering if anyone could help. I worked through what I thought was the right method and got to the answer that n=2.5 and a=6 although I thought this can't be right as you can't get a x^3 value if n=2.5. If I have got something wrong could someone please explain.


In the expansion of (1+ax)^n the coefficient of x is 15 and the coefficient of x^2 is equal to the coefficient of x^3.
Find the value of a and n.

Can you post your working as per forum rules please?

Hi, so I recently came across this question regarding Binomial expansion (attached below) and I was wondering if anyone could help. I worked through what I thought was the right method and got to the answer that n=2.5 and a=6 although I thought this can't be right as you can't get a x^3 value if n=2.5. If I have got something wrong could someone please explain.


In the expansion of (1+ax)^n the coefficient of x is 15 and the coefficient of x^2 is equal to the coefficient of x^3.
Find the value of a and n.

I guess the problem is more about your understanding of the (infinite) binomial expansion, rather than ploughing through the algebra to get the answer. Just assume that you can do the thing youre worried about so
1 + ax + bx^2 + cx^3 + ... = (1+6x)^(5/2)
So you have a fractional exponent on the right but the usual power series on the left. So square both sides
(1 + ax + bx^2 + cx^3 + ...)^2 = (1+6x)^5
and expand and equate coefficients to get a, b, c, .... So you can represent a binomial with a fractional exponent as an (infinite) power series and solving for a, b, c ... gives the usual values in your textbook.