Topic 6: Geometric Sequence and Sums

Geometric Sequence

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

For example;

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")

For 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, ... }

But be careful, r should not be 0:

When r=0, we get the sequence { a, 0, 0, ... } which is not geometric

Rule

We can also calculate any term using the Rule:

x_n = ar^(n-1)

We use "n-1" because ar⁰ is for the 1st term

For 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:

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:

For 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

Why "Geometric" Sequence?

Because it is like increasing the dimensions in geometry:

Geometric Sequences are sometimes called Geometric Progressions.

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 ar^k, 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

The formula is easy to use, just "plug in" the values of a, r and n
For 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)

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.

Using the Formula

Let's see the formula in action:

For example: Grains of Rice on a Chess Board

When we place rice on a chess board:

• 1 grain on the first square,
• 2 grains on the second square,
• 4 grains on the third and so on,
• ...

doubling the grains of rice on each square.

how many grains of rice in total?

So we have:

• a = 1 (the first term)
• r = 2 (doubles each time)
• n = 64 (64 squares on a chess board)

So:

Becomes:

= (1-2⁶⁴) / (-1) = 2⁶⁴ - 1

= 18, 446, 744, 073, 709, 551, 615

Why Does the Formula Work?

Let's see why the formula works, because we get to use an interesting "trick" which is worth knowing.

First, call the whole sum "S":

S = a + ar + ar² + ... + ar^(n-2) + ar^(n-1)

Next, multiply S by r:

S × r = ar + ar² + ar³ + ... + ar^(n-1) + arⁿ

Notice that S and S × r are similar?

Now subtract them!

All the terms in the middle neatly cancel out. 

By subtracting S × r from S we get a simple result:

S − S × r = a − arⁿ

Let's rearrange it to find S:

Factor out S and a: 

S(1− r) = a(1− rⁿ) 

Divide by (1-r):

S = a(1−rⁿ)/(1−r)

Which is our formula: