Geometric Sequences & Series (Edexcel IGCSE Further Pure Maths): Revision Note

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, $\mathit{a}$, is 2

    • The common ratio, $\mathit{r}$, 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, $\mathit{a}$, is 1

    • The common ratio, $\mathit{r}$, 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 ${\mathit{n}}^{\mathbf{t}\mathbf{h}}$ 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 $\mathit{a}$ or $\mathit{r}$

  • 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 $\mathit{a}$ and $\mathit{r}$

  • 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 $\mathit{a}$ and $\mathit{r}$ and asked to find the value of $\mathit{n}$

  • 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 $\mathit{r}\mathbf{>}\mathbf{1}$ 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 $\mathit{a}$, $\mathit{r}$ or $\mathit{n}$

  • Substitute the information you have into the formula and solve the equation

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 $\mathit{n}$ 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 $\mathit{r}$

  • If  $\left|r\right|<1$  then the sequence converges

    • $\left|r\right|<1$  is the same as   $-1<r<1$

    • In this case the sum to infinity can be calculated

  • If  $\left|r\right|\ge 1$  then the sequence does not converge

    • $\left|r\right|\ge 1$  is the same as  $r\le -1$  or  $r\ge 1$

    • In this case the sum to infinity cannot be calculated

• If $\left|r\right|<1$, then the sum converges to a finite value given by the formula

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

• $a$ 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!