## Geometric Sequences and Sums

Table of Contents

## Sequence

A Sequence is a set of things (usually numbers) that are in order.

2, 4, 6, 8, …

## Geometric Sequences

In a Geometric Sequence, each term is found by multiplying the previous term by a constant.

1, 3, 9, 27, 81, 243, …

This sequence has a factor of 3 between each number.

Each term (except the first term) is found by multiplying the previous term by 3.

## General Form

In General, we write a Geometric Sequence like this:

**{a, ar, ar**^{2}**, ar**^{3}**, … }**

where:

- a is the first term
- r is the factor between the terms (common ratio)

{1,3,9,27,…}

The sequence starts at 1 and triples each time, so

a=1 (the first term)

r=3 (the common ratio)

And we get:

{a, ar, ar^{2}, ar^{3}, … } = {1, 1×3, 1×3^{2}, 1×3^{3}, … }

{a, ar, ar^{2}, ar^{3}, … } = {1, 3, 9, 27, … }

## r should not be 0

When r=0, we get the sequence

{a,0,0,…}

which is not geometric

## The Rule

We can also calculate any term using the Rule:

**x**_{n}** = ar**^{(n-1)}

(We use “n-1” because ar^{0} is for the 1st term)

10, 20, 40, 80, 160, 320, …

This sequence has a factor of 2 between each number.

The values of a and r are:

- a = 10 (first term)
- r = 2 (common ratio)

The Rule for any term is:

**x**_{n}** = 10 × 3**^{(n-1)}

The 4th term is:

x_{4} = 10×2^{(4-1)} = 10×2^{3} = 10×8 = 80

And the 10th term is:

x_{10} = 10×2^{(10-1)} = 10×2^{9} = 10×512= 5120

A Geometric Sequence can also have smaller and smaller values:

8, 4, 2, 1, ½, ¼, …

This sequence has a factor of ½ (a half) between each number.

Its Rule is x_{n} = 4 × (½)^{n-1}

## Why “Geometric” Sequence?

Because it is like increasing the dimensions in geometry:

r | a line is 1-dimensional and has a length of r |

r^{2}^{} | in 2 dimensions a square has an area of r^{2} |

r^{3}^{} | in 3 dimensions a cube has volume r^{3} |

… | |

r^{n} | in n dimensions a cube has volume r^{n} |

## Summing a Geometric Series

To sum a Geometric Sequence, here is a handy formula.

To sum

a + ar + ar^{2} + … + ar^{(n-1)}

Each term is ark, where k starts at 0 and goes up to n-1

Use this formula:

- a is the first term
- r is the “common ratio” between terms
- n is the number of terms

Just substitute the values of a, r and n

Sum the first 4 terms of

10, 20, 40, 80, 160, 320, …

This sequence has a factor of 2 between each number.

The values of a, r and n are:

- a = 10 (first term)
- r = 2 (common ratio)
- n = 4 (to sum the first 4 terms)

So:

Check it manually:

10 + 20 + 40 + 80 = 150

It is easier to just add them in this example, as there are only 4 terms. But imagine adding 100 terms, using the formula is much easier.

## Infinite Geometric Series

So what happens when n goes to infinity?

When r is less than 1, then r^{n} goes to 0 and we get:

NOTE:

- this does not work when r is 1 or more (or less than -1):
- r must be between (not including) -1 and 1
- r should not be 0 because we get the sequence {a,0,0,…} which isn’t geometric

Add up ALL the terms of the Geometric Sequence that halves each time:

{½, ¼, ⅛ … }

We have:

a = ½ (the first term)

r = ½ (halves each time)

And so:

Adding ½ + ¼ + ⅛ + … etc equals exactly 1.