Expression of the sum of a power series

Question

I was working on a mathematical stats problem and I don't get this part (that comes from a recursive arithmetico-geometric series):

$$U_{k} = \alpha C^{2} + \alpha^{2} C^{2} + \alpha^{3} C^{2} + ... +\alpha^{k-1} C^{2}+ \alpha ^{k} U_{0}$$

$$= \alpha C^{2} \left [ 1 + \alpha +\alpha ^{2}+ ... + \alpha ^{k-2} \right ] + \alpha ^{k}U_{0}$$

$$= \alpha C^{2} (\frac{\alpha ^{k-1}-1}{\alpha - 1}) + \alpha ^{k}U_{0}$$

My particular issue is how to go from the second line to the last one. I guess there is a closed form expression of that sum but I couldn't find it anywhere in that exact form.

Thanks.

Answer 1

The general form of a geometric series is $$a+ax+ax^2+\dots+ax^{n-1}=a(1+x+x^2+\dots+x^{n-1})=\frac{a(x^n-1)}{x-1}=\frac{a(1-x^n)}{1-x}$$

Answer 2

There is an error after the second $=$ sign; it should read $\alpha C^2(1+\alpha+\alpha^2+\cdots+\alpha^{k-2})+\alpha_kU_0$. Now, use the fact that$$1+\alpha+\alpha^2+\cdots+\alpha^{k-2}=\frac{1-\alpha^{k-1}}{1-\alpha}.$$