Geometric Sequences and Sums

A Sequence is a set of things (usually numbers) that are in order.

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

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:

The sequence starts at 1 and doubles each time, so

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:

We can also calculate any term using the Rule:

xn = ar(n-1)

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

This sequence has a factor of 3 between each number.

The values of a and r are:

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:

This sequence has a factor of 0.5 (a half) between each number.

Its Rule is xn = 4 × (0.5)n-1

Because it is like increasing the dimensions in geometry:

Geometric Sequences are sometimes called Geometric Progressions (G.P.鈥檚)

To sum these:

a + ar + ar2 + ... + ar(n-1)

(Each term is ark, where k starts at 0 and goes up to n-1)

We can use this handy formula:

a is the first term r is the "common ratio" between terms n is the number of terms

What is that funny Σ symbol? It is called Sigma Notation

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

This sequence has a factor of 3 between each number.

The values of a, r and n are:

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.

Let's see the formula in action:

On the page Binary Digits we give an example of grains of rice on a chess board. The question is asked:

When we place rice on a chess board:

... doubling the grains of rice on each square ...

... how many grains of rice in total?

So we have:

So:

Becomes:

= 1−264−1 = 264 − 1

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

Which was exactly the result we got on the Binary Digits page (thank goodness!)

And another example, this time with r less than 1:

The values of a, r and n are:

So:

Becomes:

Very close to 1.

(Question: if we continue to increase n, what happens?)

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

Notice that S and S·r are similar?

Now subtract them!

Wow! All the terms in the middle neatly cancel out. (Which is a neat trick)

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

S − S·r = a − arn

Let's rearrange it to find S:

Which is our formula (ta-da!):

So what happens when n goes to infinity?

We can use this formula:

But be careful:

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

So our infnite geometric series has a finite sum when the ratio is less than 1 (and greater than −1)

Let's bring back our previous example, and see what happens:

We have:

And so:

= ½×1½ = 1

Yes, adding 12 + 14 + 18 + ... etc equals exactly 1.

Don't believe me? Just look at this square:

By adding up 12 + 14 + 18 + ...

we end up with the whole thing!

On another page we asked "Does 0.999... equal 1?", well, let us see if we can calculate it:

We can write a recurring decimal as a sum like this:

And now we can use the formula:

Yes! 0.999... does equal 1.

So there we have it ... Geometric Sequences (and their sums) can do all sorts of amazing and powerful things.