Let's learn about the geometric sequence with some examples in A-Level Maths!
Geometric Sequences
In a Geometric Sequence each term is found by multiplying the previous term by a constant.
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")
But be careful, r should not be 0:
- When r=0, we get the sequence {a,0,0,...} which is not geometric
Example: {1,2,4,8,...}
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, ... }
The Rule
We can also calculate any term using the Rule:
Example:
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
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
To sum these:
a + ar + ar2 + ... + ar(n-1)
(Each term is ark, where k starts at 0 and goes up to n-1)
Example:
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)
Example: Add up the first 10 terms of the Geometric Sequence that halves each time: { 1/2, 1/4, 1/8, 1/16, ... }
The values of a, r and n are:
- a = ½ (the first term)
- r = ½ (halves each time)
- n = 10 (10 terms to add)
Infinite Geometric Series
So what happens when n goes to infinity?
We can use this formula:
***r must be between (but not including) −1 and 1 and r should not be 0 because the sequence {a,0,0,...} is not geometric
Drafted by Eunice (Maths)
Reference
https://www.mathsisfun.com/algebra/sequences-sums-geometric.html