How to derive the formula for a geometric series sum

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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:

a*r^⁰,a*r^¹,a*r^²,a*r^³,...,a*r^ⁿ ; where n+1 = no. of terms.

Now the sum of the above, S is:

S = a*r^⁰+a*r^¹+a*r^²+a*r^³+...+ar^ⁿ

Factorise r out in most terms in right hand side:

S = a*r^⁰+r*(a*r^⁰+a*r^¹+a*r^²+...+a*r^(^n-1))

Now note that the sum within the brackets includes all terms in our original sum S, save the last term a*r^ⁿ. This means we can substitute (S-a*r^ⁿ) for it. Therefore,

S = a*r^⁰+r*(S-a*r^ⁿ)

Now all we are left to do is make the value of the sum, S, the subject of the equation:

S = a+r*S-a*r^(ⁿ⁺¹⁾

S-r*S = a-a*r^(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)

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