Geometric Sequences

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

You are watching: 1 2 4 8 16 32 64 128

This sequence has actually a variable of 2 in between each number.

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


In General we write a Geometric Sequence prefer this:

a, ar, ar2, ar3, ...


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

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

The succession starts in ~ 1 and also doubles each time, so

a=1 (the very first term) r=2 (the "common ratio" in between terms is a doubling)

And us get:

a, ar, ar2, ar3, ...

= 1, 1×2, 1×22, 1×23, ...

= 1, 2, 4, 8, ...

But it is in careful, r must not it is in 0:

once r=0, we acquire the succession a,0,0,... I beg your pardon is not geometric

The Rule

We can likewise calculate any term utilizing the Rule:

This sequence has actually a variable of 3 between each number.

The worths of a and also r are:

a = 10 (the an initial term) r = 3 (the "common ratio")

The dominance for any 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 hatchet 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 dominion is xn = 4 × (0.5)n-1

Why "Geometric" Sequence?

Because it is like increasing the dimensions in geometry:

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

Summing a Geometric Series

To amount these:

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

(Each hatchet is ark, whereby k starts at 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 number of terms

What is that funny Σ symbol? that is dubbed Sigma Notation

(called Sigma) method "sum up"

And below and above it are presented the beginning and ending values:


It states "Sum up n where n goes indigenous 1 come 4. Answer=10

This sequence has actually a variable of 3 in between each number.

The worths of a, r and n are:

a = 10 (the first term) r = 3 (the "common ratio") n = 4 (we want to sum the very first 4 terms)




You can check it yourself:

10 + 30 + 90 + 270 = 400

And, yes, it is much easier to just add them in this example, as there are only 4 terms. Yet imagine including 50 terms ... Then the formula is much easier.

Example: grains of Rice ~ above a Chess Board


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

When we location rice top top a chess board:

1 grain on the an initial square, 2 seed on the second square, 4 grains on the 3rd and for this reason on, ...

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

... How countless grains of rice in total?

So us have:

a = 1 (the an initial 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 obtained on the Binary Digits page (thank goodness!)

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

Example: include up the an initial 10 terms of the Geometric Sequence that halves every time:

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

The values of a, r and also n are:

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




Very close come 1.

(Question: if we proceed to boost n, what happens?)

Why go the Formula Work?

Let"s watch why the formula works, because we get to usage an exciting "trick" i beg your pardon is worth knowing.

First, call the totality 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 the S and also S·r space similar?

Now subtract them!


Wow! every the state in the center neatly cancel out. (Which is a practiced trick)

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

S − S·r = a − arn

Let"s rearrange the to find S:

Factor out S
and also a:S(1−r) = a(1−rn)
Divide through (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 have to be between (but no including) −1 and also 1

and r must not it is in 0 due to the fact that the succession a,0,0,... Is not geometric

So our infnite geometric series has a finite sum once the proportion is less than 1 (and better than −1)

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

Example: include up every the terms of the Geometric Sequence the halves each time:

12, 14, 18, 116, ...

We have:

a = ½ (the an initial term) r = ½ (halves each time)

And so:


= ½×1½ = 1

Yes, adding 12 + 14 + 18 + ... etc equates to exactly 1.

Don"t believe me? simply look at this square:

By including up 12 + 14 + 18 + ...

we finish up through the entirety thing!


Recurring Decimal

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

Example: calculation 0.999...

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


And currently we can use the formula:


Yes! 0.999... does same 1.

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So over there we have actually it ... Geometric assignment (and your sums) deserve to do every sorts of impressive and powerful things.

Sequences Arithmetic Sequences and also Sums Sigma Notation Algebra Index