Geometric Sequences & Series (Leaving Cert Mathematics): Revision Notes

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Geometric Series

Introduction

Note

A geometric series is the sum of a geometric sequence.

In general :

a+ar+ar2+ar3+ar4,...a+ar+ar^2 + ar^3+ar^4, ...

The general term of any arithmetic sequence is

Note

General term of a geometric series: Page 22

Sn=a(1−rn)1−rS_n=\frac{a(1-r^n)}{1-r}

Example

Note

Find the sum of the first 10 terms in the geometric sequence 64, 32, 16,...

S10=(64)(1−(3264)10)1−(3264)=(64)(1−(12)10)12=10238\begin{align*} S_{10}&=\frac{(64)(1-(\tfrac{32}{64})^{10})}{1-(\tfrac{32}{64})} \\\\ &=\frac{(64)(1-(\tfrac{1}{2})^{10})}{\tfrac{1}{2}} \\\\ &=\tfrac{1023}{8} \end{align*}
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