Sequence is a set of things (usually numbers) that in order.
In a Geometric Sequence each term is found by multiplying the previous term by a constant.
For geometric sequences, the common ratio is r, and the first term a1 is often referred to simply as"a".
Since you get the next term by multiplying by the common ratio, the value of a2 is just ar.
The third term is a3 = r(ar) = ar2.
The fourth term is a4 = r(ar2) = ar3.
Following this pattern,
the n-th term an will have the form
an = ar(n – 1).
Examples of Geometric Sequence
Example 1:
| 2, 4, 8, 16, 32, 64, 128, 256, ... |
This sequence has a factor of 2 between each number. Each term (except the first term) is found by multiplying the previous term by 2.
In General we write a Geometric Sequence like this:
{a, ar, ar2, ar3, ... }
where:
- a is the first term, and
- r is the factor between the terms (called the "common ratio")
| 2, 4, 8, 16, 32, 64, 128, 256, ... |
The sequence starts at 1 and doubles each time, so
- a=1 (the first term)
- r=2 (the "common ratio" between terms is a doubling)
And we get :
{a, ar, ar2, ar3, ... }
= {1, 1×2, 1×22, 1×23, ... }
= {1, 2, 4, 8, ... }
But be careful, 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)
Example 2:
| 10, 30, 90, 270, 810, 2430, ... |
This sequence has a factor of 3 between each number.
The values of a and r are:
- a = 10 (the first term)
- r = 3 (the "common ratio")
The Rule for any term is:
xn = 10 × 3(n-1)
So, the 4th term is:
x4 = 10×3(4-1)
= 10×33
= 10×27
= 270
And the 10th term is:
x10 = 10×3(10-1)
= 10×39
= 10×19683
= 196830
Example 2:
A Geometric Sequence can also have smaller and smaller values:
Example:
| 4, 2, 1, 0.5, 0.25, ... |
This sequence has a factor of 0.5 (a half) between each number.
Its Rule is
xn = 4 × (0.5)n-1
Summing a Geometric Series
When we need to sum a Geometric Sequence, there 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
 | (called Sigma) means "sum up" |
And below and above it are shown the starting and ending values:
It says "Sum up n where n goes from 1 to 4.
Answer = 10
The formula is easy to use ... just "plug in" the values of a, r and n
Example of Summing a Geometric Series
Sum the first 4 terms of
| 10, 30, 90, 270, 810, 2430, ... |
This sequence has a factor of 3 between each number.
The values of a, r and n are:
- a = 10 (the first term)
- r = 3 (the "common ratio")
- n = 4 (we want to sum the first 4 terms)
So:
Becomes:
You can check it yourself:
10 + 30 + 90 + 270 = 400
And, yes, it is easier to just add them in this example, as there are only 4 terms. But imagine adding 50 terms ... then the formula is much easier.
References:
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