What is the formula to find the sum of increasing power of the same rational number? For example:
$$({2\over3})^0 + ({2\over3})^1 + ({2\over3})^2 + ({2\over3})^3 + ..... + ({2\over3})^{1000}$$ $$ = 1 + ({2\over3}) + ({2\over3})^2 + ({2\over3})^3 + ..... + ({2\over3})^{1000}$$ $$ = {1 ( 1 - ({2\over3})^{2001})\over{1-{2\over3}}}$$
I saw somewhere, but they did not explain how they got here. What is the name of the formula that allows us to get this?
PS: Sorry if it's a duplicate. Didn't know how to search it up.
It's the formula for the sum of a geometric series:
$1+r+r^2+...+r^n =\dfrac{1-r^{n+1}}{1-r} $.
This is true for any complex $r \ne 1$.
If you write it in the form
$(1-r)(1+r+r^2+...+r^n) =1-r^{n+1} $ it is true for all $r$.