Geometric Sequences

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

You are watching: 2+4+8+16+32+64

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

Each hatchet (except the an initial term) is uncovered by multiplying the previous term by 2.


In General we compose a Geometric Sequence like this:

a, ar, ar2, ar3, ...


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

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

The sequence starts at 1 and doubles every time, so

a=1 (the very 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 should not be 0:

when r=0, we gain the sequence a,0,0,... Which is not geometric

The Rule

We can also calculate any term utilizing the Rule:

This sequence has a factor of 3 in between each number.

The worths of a and also r are:

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

The preeminence 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 ax is:

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

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

Its preeminence is xn = 4 × (0.5)n-1

Why "Geometric" Sequence?

Because that is like increasing the dimensions in geometry:

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

Summing a Geometric Series

To amount these:

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

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

We can use this comfortable formula:

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

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

(called Sigma) means "sum up"

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


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

This sequence has a variable of 3 between each number.

The worths of a, r and n are:

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




You can inspect it yourself:

10 + 30 + 90 + 270 = 400

And, yes, the is much easier to just add them in this example, as there are only 4 terms. But imagine including 50 state ... Climate the formula is lot 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 concern is asked:

When we location rice on a chess board:

1 grain on the first square, 2 grains on the 2nd square, 4 seed on the third and so on, ...

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

... How numerous grains the rice in total?

So us have:

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




= 1−264−1 = 264 − 1

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

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

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

Example: include up the first 10 terms of the Geometric Sequence the halves each time:

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

The values of a, r and also n are:

a = ½ (the an initial term) r = ½ (halves every time) n = 10 (10 terms to add)




Very close come 1.

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

Why go the Formula Work?

Let"s check out why the formula works, because we gain to use an interesting "trick" which is precious knowing.

First, contact 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 S·r space similar?

Now subtract them!


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

By individually S·r indigenous S we obtain a straightforward result:

S − S·r = a − arn

Let"s rearrange it to uncover 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 our formula (ta-da!):

Infinite Geometric Series

So what happens once n goes come infinity?

We deserve to use this formula:


But be careful:

r should be in between (but not including) −1 and 1

and r have to not it is in 0 since the sequence a,0,0,... Is no geometric

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

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

Example: add up all the terms of the Geometric Sequence the halves every time:

12, 14, 18, 116, ...

We have:

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

And so:


= ½×1½ = 1

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

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

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

we finish up through the totality 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 have the right to write a recurring decimal as a sum favor this:


And currently we deserve to use the formula:


Yes! 0.999... does same 1.

See more: Under Which Conditions Of Temperature And Pressure Does A Real Gas Behavior

So over there we have it ... Geometric assignment (and your sums) can do all sorts of impressive and powerful things.

Sequences Arithmetic Sequences and Sums Sigma Notation Algebra Index