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) 

ZA
Answered by Zeeshan A. Maths tutor

7425 Views

See similar Maths GCSE tutors

Related Maths GCSE answers

All answers ▸

Find the co-ordinates of the turning point of the line with equation y = x^2 + ax + b that passes through (1, 47) and (2, 60)


2 5/3 + 2 8/9


Solve the equation 2X^2 + 5X + 2 = 0 stating clearly the number of roots


161/5 x 26x =


We're here to help

contact us iconContact ustelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

MyTutor is part of the IXL family of brands:

© 2025 by IXL Learning