How do I find the two variables a and n?

In the expansion of ${\left(1 + a x\right)}^{n}$ the first three terms are 1+(5/3)x + (10/9) ${x}^{2}$ what is a and what is n?

Jun 18, 2018

$a = \frac{1}{3} , n = 5$, i.e. the expression is ${\left(1 + \frac{1}{3} x\right)}^{5}$

Explanation:

To solve this you need to know the formula for the binomial expansion of ${\left(1 + a x\right)}^{n}$ 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:
$1 + n \left(a x\right) + \frac{n \left(n - 1\right)}{2} {\left(a x\right)}^{2}$
= (1+(5/3)x+(10/9)x^2
Therefore
$a n = \frac{5}{3}$
${a}^{2} \frac{n \left(n - 1\right)}{2} = \frac{10}{9}$
or ${a}^{2} \left(n \left(n - 1\right)\right) = \frac{20}{9}$

For this to work properly we notice that a=1/3, since n must be a natural number, therefore n=5
Test: ${\left(\frac{1}{3}\right)}^{2} \left(5 \cdot 4\right) = \frac{20}{9}$ Check.

Our expression, then, is ${\left(1 + \frac{x}{3}\right)}^{5}$