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?

1 Answer
Jun 18, 2018

#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#