Geometric Sequences

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

You are watching: 16+8+4+2+1


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

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

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In General we write a Geometric Sequence like this:

a, ar, ar2, ar3, ...

where:

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


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

The succession 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 it is in careful, r should not it is in 0:

once r=0, we get the sequence a,0,0,... Which is no geometric

The Rule

We can additionally calculate any term making use of 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 dominion for any kind 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 aspect of 0.5 (a half) between each number.

Its preeminence is xn = 4 × (0.5)n-1


Why "Geometric" Sequence?

Because that is like boosting the dimensions in geometry:

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


Summing a Geometric Series

To amount these:

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

(Each term is ark, whereby k starts at 0 and also goes as much as n-1)

We deserve to use this comfortable formula:

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


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

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(called Sigma) method "sum up"

And listed below and over it are displayed the starting and ending values:

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It claims "Sum up n where n goes from 1 to 4. Answer=10


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

The worths of a, r and also n are:

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

So:

Becomes:

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You can examine it yourself:

10 + 30 + 90 + 270 = 400

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


Example: seed of Rice top top a Chess Board

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On the web page Binary Digits us give an example of grains of rice ~ above a chess board. The inquiry is asked:

When we place rice top top a chess board:

1 serial on the first square, 2 grains on the second square, 4 grains on the third and therefore on, ...

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

... How countless grains that rice in total?

So we have:

a = 1 (the an initial term) r = 2 (doubles every time) n = 64 (64 squares ~ above a chess board)

So:

Becomes:

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= 1−264−1 = 264 − 1

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

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


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


Example: include up the very first 10 regards to the Geometric Sequence the halves each time:

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

The worths of a, r and also n are:

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

So:

Becomes:

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Very close come 1.

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


Why go the Formula Work?

Let"s watch why the formula works, due to the fact that we obtain to usage an amazing "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 that S and also S·r room similar?

Now subtract them!

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

By subtracting S·r native S we acquire a an easy 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 (ta-da!):

Infinite Geometric Series

So what happens once n goes come infinity?

We can use this formula:

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But be careful:


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

and r have to not be 0 since the sequence a,0,0,... Is no geometric


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

Let"s bring earlier our vault example, and 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:

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= ½×1½ = 1

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


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

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

we end up through the totality thing!

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Recurring Decimal

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


Example: calculation 0.999...

We can write a recurring decimal together a sum favor this:

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And now we have the right to use the formula:

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Yes! 0.999... does equal 1.

See more: Top 10 Fastest Land Animals In The World ( How Fast Can A Giraffe Run ?


So over there we have it ... Geometric assignment (and your sums) deserve to do all sorts of amazing and an effective things.


Sequences Arithmetic Sequences and also Sums Sigma Notation Algebra Index