A geometric series has first term a a\,a and common ratio rrr, where r≠1r \neq 1r=1. The sum of the first n n\,n terms of the series is SnS_nSn.
By writing down expressions for Sn S_n\,Sn and for rSn rS_n\,rSn and subtracting one from the other, prove that
Sn=a(1−rn)1−rS_n = \dfrac{a(1 - r^n)}{1 - r}Sn=1−ra(1−rn)
308 exam-style questions on OCR A Level Maths 1.4 Sequences and Series, covering 1.4.1 Binomial expansion for positive integer n, 1.4.2 Link to binomial probabilities, 1.4.3 Binomial expansion for rational n (A-level only), 1.4.4 Validity of the expansion (A-level only), 1.4.5 Sequences (A-level only), 1.4.6 Increasing, decreasing and periodic sequences (A-level only), 1.4.7 Sigma notation (A-level only), 1.4.8 Arithmetic sequences and series (A-level only), 1.4.9 Geometric sequences and series (A-level only), 1.4.10 Sum to infinity of a geometric series (A-level only), and 1.4.11 Modelling with sequences and series (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.