Geometric Sequences & Series (Edexcel IGCSE Further Maths)

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Amber

Expertise

Maths

Geometric Sequences

What is a geometric sequence?

In a geometric sequence, there is a common ratio between consecutive terms in the sequence
- This means each term is multiplied by a common ratio to get the next term
The first term of the sequence is denoted by $a$
The common ratio is denoted by $r$
- For example, 2, 6, 18, 54, 162, … is a sequence with the rule ‘start at two and multiply each number by three’
  - The first term, $a$ , is 2
  - The common ratio, $r$ , is 3
A geometric sequence can be
- increasing (r > 1), or
- decreasing (0 < r < 1)
If the common ratio is a negative number the terms will alternate between positive and negative values
- For example, 1, -4, 16, -64, 256, … is a sequence with the rule ‘start at one and multiply each number by negative four’
  - The first term, $a$ , is 1
  - The common ratio, $r$ , is -4
Terms in a geometric sequence can be referred to
- by the letter $u$ with
- a subscript corresponding to its place in the sequence
  - e.g. $u_{1} = a$ is the first term, $u_{9}$ is the ninth term, $u_{n}$ is the $n$ ^th term, etc.

Geometric Series

What is a geometric series?

When the terms of a geometric sequence are added together, that is known as a geometric series
- The terms (1st term, 2nd term, 3rd term, etc.) are exactly the same in the sequence and series
- But with the series we're most interested in what happens when the terms are added together

How do I find a term in a geometric series?

The $n^{t h}$ term formula for a geometric series is

$u_{n} = a r^{n - 1}$

- Where $a$ is the first term, and $r$ is the common ratio
- This is not given on the exam formula sheet, so make sure you know it
The formula allows you to find any term in the geometric series
- Enter the values of $a$ , $r$ and $n$ and calculate the value of $u_{n}$
Sometimes you will be given a term and asked to find $a$ or $r$
- Substitute the information you have into the formula and solve the equation
Sometimes you will be given two or more consecutive terms and asked to find both $a$ and $r$
- Find the common ratio by dividing a term by the one immediately before it
  - $r = \frac{u_{n + 1}}{u_{n}}$
- Substitute this and one of the terms into the formula to find the first term
Sometimes you may be given a term along with $a$ and $r$ and asked to find the value of $n$
- You can solve this using logarithms on your calculator

How do I find the sum of a geometric series?

A geometric series is the sum of the terms in a geometric sequence
- For the geometric sequence 2, 6, 18, 54, … the geometric series is 2 + 6 + 18 + 54 + …
Use the following formula to find the sum of the first n terms of a geometric series:

$S_{n} = \frac{a (1 - r^{n})}{1 - r}$

- - $a$ is the first term
  - $r$ is the common ratio
- The formula is given on the exam formula sheet
  - So you don't need to remember it
  - But you do need to know how to use it!
If $r > 1$ the following rearrangement of the formula might be more convenient:

$S_{n} = \frac{a (r^{n} - 1)}{r - 1}$

- This version is not on the formula sheet
- The formula sheet version will always work as well
A question will often give you the sum of a certain number of terms and ask you to find the values of $a$ , $r$ or $n$
- Substitute the information you have into the formula and solve the equation

Exam Tip

The formula for the sum of a geometric series is on the exam formula sheet
- But the n^th term formula is not on the formula sheet
You will sometimes need to use logarithms to answer geometric series questions
- Make sure you are confident doing this
- And know how your calculator handles logarithms

Worked example

The sixth term of a geometric series is 486 and the seventh term is 1458.

Find

(a)

the common ratio of the series

Find the common ratio by dividing a term by the term immediately before it
Here

r = \frac{7^{th} term}{6^{th} term}

r = \frac{1458}{486}

r = 3

(b)

the first term of the series.

Use the n^th term formula

u_{n} = a r^{n - 1}

For the 6^th term, we know

u_{6} = 486

r = 3

and

n = 6

\begin{array}{rcl} 486 & = & a \times 3^{(6 - 5)} \\ 486 & = & a \times 3^{5} \\ 486 & = & 243 a \\ a & = & \frac{486}{243} \end{array}

a = 2

Worked example

The first term of a geometric series is 25, and the common ratio is 0.8.

Find the value of the fifth term, as well as the sum of the first 5 terms.

To find the fifth term, use the n^th term formula $u_{n} = a r^{n - 1}$
Here $n = 5$ , $a = 25$ and $r = 0.8$

$\begin{array}{rcl} fifth term & = & 25 \times 0 . 8^{(5 - 1)} \\ = & 25 \times 0 . 8^{4} \end{array}$

$fifth term = 10.24$

To find the sum of the first 5 terms, use the sum of a geometric series formula $S_{n} = \frac{a (1 - r^{n})}{1 - r}$
Again $n = 5$ , $a = 25$ and $r = 0.8$

$S_{5} = \frac{25 (1 - 0 . 8^{5})}{1 - 0.8}$

Use your calculator to work this out

$S_{5} = 84.04$

Sum to Infinity

What is the sum to infinity of a geometric series?

The sum to infinity is the sum of all the terms in a geometric series
- $u_{1} + u_{2} + u_{3} + . . .$ 'all the way to infinity'
As $n$ increases the terms of a geometric series may
- move further away from zero
  - if $r > 1$ or $r < - 1$
- stay the same distance away from zero
  - if $r = 1$ or $r = - 1$
- get closer and closer to zero
  - if $- 1 < r < 1$
If the terms are getting closer to zero then the series is said to converge

- This means that the sum of the series will approach a finite 'limiting value'
- As $n$ increases, the sum of the terms will get closer and closer to the limiting value
- The limiting value is the sum to infinity of the series
  - It is denoted by $S_{\infty}$

How do I calculate the sum to infinity?

First you need to consider the value of $r$
- If $|r| < 1$ then the sequence converges
  - $|r| < 1$ is the same as $- 1 < r < 1$
  - In this case the sum to infinity can be calculated
- If $|r| \geq 1$ then the sequence does not converge
  - $|r| \geq 1$ is the same as $r \leq - 1$ or $r \geq 1$
  - In this case the sum to infinity cannot be calculated
If $| r | < 1$ , then the sum converges to a finite value given by the formula

$S_{\infty} = \frac{a}{1 - r}, |r| < 1$

- - $a$ is the first term
  - $r$ is the common ratio
- The formula is given on the exam formula sheet
  - So you don't need to remember it
  - But you do need to know how to use it!

Exam Tip

Always check the $|r| < 1$ condition before calculating a sum to infinity
- Marks may depend on showing this in your working

Worked example

The first three terms of a geometric sequence are $6, 2, \frac{2}{3}$ . Show that the sum to infinity of the series exists, and then find the sum to infinity.

To show the sum to infinity exists we need to know the value of the common ratio $r$

We can find this by dividing a term by the term immediately before it

$r = \frac{2}{6} = \frac{1}{3}$

This satisfies $|r| < 1$ , so the series converges and the sum to infinity exists

$| r | < 1$ , so the series converges

Now use the sum to infinity formula $S_{\infty} = \frac{a}{1 - r}$

Here $a = 6$ and $r = \frac{1}{3}$

$\begin{array}{rcl} S_{\infty} & = & \frac{6}{1 - \frac{1}{3}} \\ = & \frac{6}{(\frac{2}{3})} \\ = & 6 \times \frac{3}{2} \end{array}$

$S_{\infty} = 9$

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Geometric Sequences & Series (Edexcel IGCSE Further Maths)

Revision Note

What is a geometric sequence?

What is a geometric series?

How do I find a term in a geometric series?

How do I find the sum of a geometric series?

What is the sum to infinity of a geometric series?

How do I calculate the sum to infinity?

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Author: Amber