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)
