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, ar**^{2}**, ar**^{3}**, ... }**

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, ar^{2}, ar^{3}, ... }

= {1, 1×2, 1×2^{2}, 1×2^{3}, ... }

= {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:

x_{n} = 10 × 3^{(n-1)}

So, the 4th term is:

x_{4} = 10×3^{(4-1)} = 10×3^{3} = 10×27 = 270

And the 10th term is:

x_{10 }= 10×3^{(10-1)} = 10×3^{9} = 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 x_{n} = 4 × (0.5)^{n-1}

**Summing a Geometric Series **

To sum these:

a + ar + ar^{2} + ... + ar^{(n-1)}

(Each term is ar^{k}, 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