Geometric Series

Maths: Edexcel A Level Pure Maths, topic 4.2.8

Geometric Series

The sum of the terms in a geometric sequence is called a geometric series.

Series

Series

The sum of the terms of a sequence is called a series.
For the geometric sequence defined by $a_n = 5\times 2^n$ , the first four terms are:
$5,10,20,40$
The series is then given by:
$5 +10+20+40$
The sum of this series is 75.

General formula

General formula

We can write the sum of the first $n$ terms of a geometric series with common ratio $r$ as:
$S_n = a_1 + ra_1+ ...+ r^{n-1}a_1$
Multiplying both sides of the equation by $r$ we have:
$rS_n = ra_1 + r^2a_1+...+ r^na_1$

General formula

General formula

Subtracting this from the original equation for $S_n$ , almost all of the terms cancel out:
$S_n - rS_n = a_1 - r^na_1$
Taking out a factor of $S_n$ and dividing by $1-r$ , we have the general equation for the sum of a geometric series:
$\large S_n = \large a_1 \frac{\large(1-r^n)}{\large(1-r)},\; r\ne 1$

Test yourself on Geometric Series

Which term best describes the following:
$4, 16, 64, 256,...$
Which term best describes the following:
$66+33+16.5+8.25+...$
Which term best describes the following:
$33+27+21+15+...$

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