How do I simplify an undefined sum of powers?

Question

The problem:
There are 456 million license plate combinations. 900 thousand cars are sold "every year" (the first year) with an annual increment of 5% over the new number (the year before). How many years would it take to use every license plate number?
The solution I found:
1st year: 900k cars, 2nd year: 900k+(1stYear*5%) cars, 3rd year: 900k+(2ndYear*5%) cars...
$\sum_{i=1}^{years} 900k\times1,05^{(i-1)} = 456m$
Solving a bit you get this:
$years\times900k+(1,05^0+1,05^1+1,05^2...+1,05^{x-1})=456m$
My problem:
How can I solve that? I mean, how can I factorize/simplify the powers expression? Is my approach even accurate?

Answer 1

HINT

Recall that for the geometric series

$$\ S = \sum_{j=0}^n ar^j = \frac{ar^{n+1}-a}{r-1}$$