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, ar2, ar3, … }
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, ar2, ar3, … } = {1, 1×3, 1×32, 1×33, … }
{a, ar, ar2, ar3, … } = {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:
xn = ar(n-1)
(We use “n-1” because ar0 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:
xn = 10 × 3(n-1)
The 4th term is:
x4 = 10×2(4-1) = 10×23 = 10×8 = 80
And the 10th term is:
x10 = 10×2(10-1) = 10×29 = 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 xn = 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 |
| r2 | in 2 dimensions a square has an area of r2 |
| r3 | in 3 dimensions a cube has volume r3 |
| … | |
| rn | in n dimensions a cube has volume rn |
Summing a Geometric Series
To sum a Geometric Sequence, here is a handy formula.
To sum
a + ar + ar2 + … + 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 rn 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.