Geometric Series

A LevelAQAEdexcelOCR

Geometric Series

A series is a sequence where the goal is to add all the terms together. We will study arithmetic series and geometric series.

Recall: Notation from Sequences:

$a$ is first term.

$r$ is the ratio, the amount we multiply by each time.

$n$ is the number of terms in the series.

As for arithmetic series, we can add together all the terms in a geometric series.

Make sure you are happy with the following topics before continuing.

Sequences

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Adding the Terms

The sum of a geometric series with $n$ terms is:

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

We can prove this result:

$S_{n}=a+ar+ar^{2}+…+ar^{n-1}$

$rS_{n}=ar+ar^{2}+ar^{3}+…+ar^{n}$

$S_{n}-rS_{n}=a-ar^{n}$

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

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

Sum to Infinity

If $|r|<1$ in a geometric progression, then we can sum up an infinite number of terms and still end with a finite answer. This is called the sum to infinity. The sum to infinity is given by:

$S_{infty}=dfrac{a}{1-r}$

Series that have a sum to infinity are convergent.

Series that do not have a sum to infinity are divergent.

Example 1: Sum of a Geometric Series

What is the sum of the first ten terms of the series that begins $1,3,9,27…$ ?

[2 marks]

$a=1$

$r=3$

$n=10$

Substitute into formula:

$begin{aligned}S_{10}&=dfrac{1(1-3^{10})}{1-3}[1.2em]&=dfrac{1-3^{10}}{-2}[1.2em]&=dfrac{1-59049}{-2}[1.2em]&=dfrac{-59048}{-2}[1.2em]&=29524end{aligned}$

Example 2: Sum to Infinity

What is the sum to infinity of the geometric series that begins $1,dfrac{1}{2},dfrac{1}{4},dfrac{1}{8}…$ ?

[2 marks]

$a=1$

$r=dfrac{1}{2}$

Substitute into formula:

$begin{aligned}S_{infty}&=dfrac{a}{1-r}[1.2em]&=dfrac{1}{1-dfrac{1}{2}}[1.2em]&=dfrac{1}{dfrac{1}{2}}[1.2em]&=2end{aligned}$

Geometric Series Example Questions

Question 1: A geometric progression has first term $729$ and ratio $dfrac{2}{3}$ . Find:

i) The sum of the first 7 terms.

ii) The sum to infinity.

[4 marks]

i) $a=729$

$r=dfrac{2}{3}$

$n=7$

Substitute into formula:

$begin{aligned}S_{n}&=dfrac{a(1-r^{n})}{1-r}[1.2em]&=dfrac{729left(1-left(dfrac{2}{3}right)^{7}right)}{1-dfrac{2}{3}}[1.2em]&=dfrac{729left(1-dfrac{128}{2187}right)}{dfrac{1}{3}}[1.2em]&=3times 729left(dfrac{2059}{2187}right)[1.2em]&=2187timesdfrac{2059}{2187}[1.2em]&=2059end{aligned}$

ii) $begin{aligned}S_{infty}&=dfrac{a}{1-r}[1.2em]&=dfrac{729}{1-dfrac{2}{3}}[1.2em]&=dfrac{729}{dfrac{1}{3}}[1.2em]&=3times 729[1.2em]&=2187end{aligned}$

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Question 2: The diameter of a tree grows by a factor of $1.01$ each year. It is measured in 2009 and it has a diameter of $0.2$ m.

i) Is a sum to infinity a sensible question in this scenario?

ii) If a tree of this type is planted every year up to 2030, starting in 2009, what is the sum of all of the diameters?

[4 marks]

i) A sum to infinity is not sensible for two reasons:

The tree will not live forever.

$r=1.01$ so $|r|>1$ so there is no finite sum to infinity.

ii) $a=0.2$

$r=1.01$

$n=22$

Substitute into formula:

$begin{aligned}S_{n}&=dfrac{a(1-r^{n})}{1-r}[1.2em]&=dfrac{0.2(1-1.01^{21})}{1-1.01}[1.2em]&=4.65end{aligned}$ m

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Question 3: The sum to infinity of a series is $390$ . The first term is $2$ . Find $r$ .

[2 marks]

$S_{infty}=dfrac{a}{1-r}$

$a=2$

$390=dfrac{2}{1-r}$

$1-r=dfrac{2}{390}$

$r=1-dfrac{2}{390}$

$r=dfrac{388}{390}$

$r=dfrac{194}{195}$

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Question 4: On one chessboard is placed $£1$ on the first square, $£2$ on the second square, $£3$ on the third square, etc. On another chessboard is placed $1$ p on the first square, $2$ p on the second square, $4$ p on the third square, etc. Which chessboard has more money on it overall?

Hint: A chessboard has 64 squares.

[6 marks]

The first option is an arithmetic series:

$a=1$

$d=1$

$n=64$

Substitute into formula:

$begin{aligned}S_{n}&=dfrac{n}{2}(2a+(n-1)d)[1.2em]&=dfrac{64}{2}(2times 1+(64-1)1)[1.2em]&=32(2+63)[1.2em]&=32times 65[1.2em]&=£2080end{aligned}$

The second option is a geometric series:

$a=0.01$

$r=2$

$n=64$

Substitute into formula:

$begin{aligned}S_{n}&=dfrac{a(1-r^{n})}{1-r}[1.2em]&=dfrac{0.01(1-2^{64})}{1-2}[1.2em]&=£1.84times 10^{17}end{aligned}$

So the second chessboard is far more valuable.

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Additional Resources

Exam Tips Cheat Sheet

A Level

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Formula Booklet

A Level

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Specification Points Covered

D5 – Understand and work with geometric sequences and series including the formulae for the $n$ th term and the sum of a finite geometric series; the sum to infinity of a convergent geometric series, including the use of $|r| < 1$ ; modulus notation

Geometric Series Worksheet and Example Questions

Sequences and Series

Exam Questions Mark Scheme

Revise

Try it yourself

Geometric Series

Geometric Series

A Level Maths Predicted Papers 2025

A-A* A Level Maths Practice Papers

Adding the Terms

Sum to Infinity

Example 1: Sum of a Geometric Series

Example 2: Sum to Infinity

Geometric Series Example Questions