Question

How do I find the two variables a and n?

In the expansion of `(1+ax)^n` the first three terms are 1+(5/3)x + (10/9) `x^2` what is a and what is n?

Answer 1

`a=1/3, n=5`, i.e. the expression is `(1+1/3x)^5`

Explanation:

To solve this you need to know the formula for the binomial expansion of `(1+ax)^n` which is:
`1+ n(ax)+(n(n-1))/(2!)(ax)^2+(n(n-1)(n-2))/(3!)(ax)^3 + ....`

In our situation we only need the first 3 terms:
`1+ n(ax)+(n(n-1))/2(ax)^2`
= `(1+(5/3)x+(10/9)x^2`
Therefore
`an=5/3`
`a^2(n(n-1))/2=10/9`
or `a^2(n(n-1))=20/9`

For this to work properly we notice that a=1/3, since n must be a natural number, therefore n=5
Test: `(1/3)^2(5*4)=20/9` Check.

Our expression, then, is `(1+x/3)^5`