How to derive the formula for a geometric series sum

Any geometric series is defined by an initial term a, and a common ratio b. This means that we start with a and multiply by b to get each next term.And so the general geometric series is written:ar^⁰,ar^¹,ar^²,ar^³,...,ar^ⁿ ; where n+1 = no. of terms.Now the sum of the above, S is:S = ar^⁰+ar^¹+ar^²+ar^³+...+ar^ⁿFactorise r out in most terms in right hand side:S = ar^⁰+r*(ar^⁰+ar^¹+ar^²+...+ar^(^n-1))Now note that the sum within the brackets includes all terms in our original sum S, save the last term ar^ⁿ. This means we can substitute (S-ar^ⁿ) for it. Therefore,S = ar^⁰+r(S-ar^ⁿ)Now all we are left to do is make the value of the sum, S, the subject of the equation:S = a+rS-ar^(ⁿ⁺¹⁾S-rS = a-ar^(ⁿ⁺¹⁾S(1-r) = a*(1-r*(ⁿ⁺¹⁾); factorize S on LHS, and a on RHS.S = a*(1-r*(ⁿ⁺¹⁾)/(1-r)