Geometric Sequences

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

You are watching: 1 2 4 8 16 pattern name


This sequence has a element of 2 between each number.

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

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In General we create a Geometric Sequence choose this:

a, ar, ar2, ar3, ...

where:

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


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

The sequence starts in ~ 1 and also doubles every time, so

a=1 (the 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 need to not be 0:

once r=0, we obtain the succession a,0,0,... I beg your pardon is no geometric

The Rule

We can also calculate any term making use of the Rule:


This sequence has a factor of 3 between each number.

The values of a and also r are:

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

The ascendancy for any term is:

xn = 10 × 3(n-1)

So, the 4th hatchet 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 actually a element of 0.5 (a half) between each number.

Its ascendancy is xn = 4 × (0.5)n-1


Why "Geometric" Sequence?

Because that is like boosting the size in geometry:

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a line is 1-dimensional and also 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 have the right to have 4 and an ext dimensions in mathematics).


Summing a Geometric Series

To sum these:

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

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

We deserve to 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? that is called Sigma Notation

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(called Sigma) method "sum up"

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

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


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

The values of a, r and also n are:

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

So:

Becomes:

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You can check it yourself:

10 + 30 + 90 + 270 = 400

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


Example: grains of Rice top top a Chess Board

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

When we place rice on a chess board:

1 serial on the very first square, 2 seed on the 2nd square, 4 seed on the third and for this reason on, ...

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

... How countless grains the rice in total?

So we have:

a = 1 (the very first term) r = 2 (doubles each 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 specifically the result we acquired on the Binary Digits page (thank goodness!)


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


Example: add up the an initial 10 terms of the Geometric Sequence that halves each time:

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

The values of a, r and n are:

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

So:

Becomes:

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Very close come 1.

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


Why does the Formula Work?

Let"s watch why the formula works, since we get to usage an interesting "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 are similar?

Now subtract them!

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

By individually S·r from S we get a an easy result:


S − S·r = a − arn


Let"s rearrange that to find S:


Factor out 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 as soon as n goes to infinity?

We can use this formula:

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But be careful:


r need to be in between (but not including) −1 and also 1

and r should not be 0 because the sequence a,0,0,... Is not geometric


So ours infnite geometric collection has a finite sum once the proportion is much less than 1 (and greater than −1)

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


Example: add up all 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 every 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 adding up 12 + 14 + 18 + ...

we end up v the whole thing!

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Recurring Decimal

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


Example: calculation 0.999...

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

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And currently we deserve to use the formula:

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Yes! 0.999... does equal 1.

See more: Muscles Are Only Able To Pull They Never Push, Your Muscles (For Kids)


So over there we have it ... Geometric order (and your sums) have the right to do all sorts of impressive and an effective things.


Sequences Arithmetic Sequences and Sums Sigma Notation Algebra Index