Geometric Sequences

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

You are watching: What is the common ratio of the sequence below?

This sequence has actually a element of 2 between each number.

Each ax (except the first term) is discovered by multiplying the previous ax by 2.


In General we write a Geometric Sequence like this:

a, ar, ar2, ar3, ...


a is the very first term, and also r is the factor in between the state (called the \"common ratio\")

Example: 1,2,4,8,...

The sequence starts at 1 and also doubles every time, so

a=1 (the first term) r=2 (the \"common ratio\" in 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 need to not be 0:

as soon as r=0, we get the succession a,0,0,... I m sorry is not geometric

The Rule

We can likewise calculate any term utilizing the Rule:

This sequence has a element of 3 between each number.

The values of a and also r are:

a = 10 (the first term) r = 3 (the \"common ratio\")

The ascendancy for any type of term is:

xn = 10 × 3(n-1)

So, the 4th ax is:

x4 = 10×3(4-1) = 10×33 = 10×27 = 270

And the 10th ax is:

x10 = 10×3(10-1) = 10×39 = 10×19683 = 196830

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

Its ascendancy is xn = 4 × (0.5)n-1

Why \"Geometric\" Sequence?

Because the is like raising the size in geometry:

a heat is 1-dimensional and has a length of r
in 2 size a square has an area of r2
in 3 size a cube has volume r3
etc (yes we can have 4 and an ext dimensions in mathematics).

Summing a Geometric Series

To sum these:

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

(Each term is ark, wherein k starts in ~ 0 and goes up to n-1)

We can use this comfortable formula:

a is the very first term r is the \"common ratio\" in between terms n is the variety of terms

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

(called Sigma) way \"sum up\"

And below and over it are shown the beginning and ending values:


It claims \"Sum up n wherein n goes from 1 come 4. Answer=10

This sequence has a aspect of 3 between each number.

The worths of a, r and n are:

a = 10 (the first term) r = 3 (the \"common ratio\") n = 4 (we desire to amount the first 4 terms)




You can inspect it yourself:

10 + 30 + 90 + 270 = 400

And, yes, that is less complicated to just include them in this example, together there are only 4 terms. But imagine adding 50 state ... Then the formula is much easier.

Example: seed of Rice top top a Chess Board


On the page Binary Digits we give an instance of seed of rice on a chess board. The inquiry is asked:

When we location rice ~ above a chess board:

1 serial on the very first square, 2 grains on the 2nd square, 4 seed on the 3rd and so on, ...

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

... How countless grains the rice in total?

So us have:

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




= 1−264−1 = 264 − 1

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

Which was precisely the result we acquired on the Binary Digits page (thank goodness!)

And an additional example, this time v r less than 1:

Example: include up the an initial 10 regards to the Geometric Sequence that halves every time:

1/2, 1/4, 1/8, 1/16, ...

The worths of a, r and n are:

a = ½ (the very first term) r = ½ (halves every time) n = 10 (10 state to add)




Very close come 1.

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

Why does the Formula Work?

Let\"s check out why the formula works, due to the fact that we obtain to use an exciting \"trick\" i m sorry is worth knowing.

First, speak to the whole sum \"S\":S= a + ar + ar2 + ... + ar(n−2)+ ar(n−1)
Next, main point S by r:S·r= ar + ar2 + ar3 + ... + ar(n−1) + arn

Notice that S and S·r space similar?

Now subtract them!


Wow! every the state in the middle neatly release out. (Which is a succinct trick)

By subtracting S·r from S we gain a an easy result:

S − S·r = a − arn

Let\"s rearrange it to find S:

Factor the end S
and also a:S(1−r) = a(1−rn)
Divide by (1−r):S = a(1−rn)(1−r)

Which is ours formula (ta-da!):

Infinite Geometric Series

So what happens once n goes to infinity?

We deserve to use this formula:


But be careful:

r should be in between (but no including) −1 and 1

and r should not it is in 0 because the sequence a,0,0,... Is not geometric

So our infnite geometric collection has a finite sum once the ratio is less than 1 (and higher than −1)

Let\"s bring ago our previous example, and see what happens:

Example: add up every the regards to the Geometric Sequence that halves each time:

12, 14, 18, 116, ...

We have:

a = ½ (the very first term) r = ½ (halves every time)

And so:


= ½×1½ = 1

Yes, including 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 v the whole thing!


Recurring Decimal

On an additional page us asked \"Does 0.999... Equal 1?\", well, let united state see if we have the right to calculate it:

Example: calculate 0.999...

We deserve to write a recurring decimal together a sum choose this:


And now we have the right to use the formula:


Yes! 0.999... does same 1.

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So over there we have it ... Geometric assignment (and their sums) have the right to do every sorts of exceptional and an effective things.

Sequences Arithmetic Sequences and also Sums Sigma Notation Algebra Index