Geometric Sequences

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

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


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

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

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In General we compose a Geometric Sequence favor this:

a, ar, ar2, ar3, ...

where:

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


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

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

a=1 (the an initial 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 be careful, r need to not it is in 0:

once 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 r are:

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

The dominion for any type of 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 a variable of 0.5 (a half) between each number.

Its dominance is xn = 4 × (0.5)n-1


Why "Geometric" Sequence?

Because that is like enhancing the size in geometry:

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a line is 1-dimensional and also has a size of r
in 2 size a square has an area of r2
in 3 dimensions a cube has actually volume r3
etc (yes we deserve 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, wherein k starts at 0 and goes up to n-1)

We can 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? it is referred to as Sigma Notation

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(called Sigma) means "sum up"

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

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It claims "Sum increase n whereby n goes native 1 to 4. Answer=10


This sequence has actually a element of 3 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 very first 4 terms)

So:

Becomes:

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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. However imagine adding 50 state ... Then the formula is much easier.


Example: grains of Rice ~ above a Chess Board

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

When we location rice top top a chess board:

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

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

... How many grains of rice in total?

So us 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 precisely the an outcome we gained on the Binary Digits web page (thank goodness!)


And one more example, this time through 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 worths of a, r and n are:

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

So:

Becomes:

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Very close come 1.

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


Why go the Formula Work?

Let"s see why the formula works, due to the fact that we obtain to use an exciting "trick" i m sorry is precious 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 the S and S·r are similar?

Now subtract them!

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

By subtracting S·r native S we acquire a straightforward result:


S − S·r = a − arn


Let"s rearrange it to discover 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 when n goes to infinity?

We deserve to use this formula:

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But be careful:


r have to be between (but no including) −1 and also 1

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


So ours infnite geometric collection has a finite sum as soon as the ratio is less than 1 (and greater than −1)

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


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

12, 14, 18, 116, ...

We have:

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

And so:

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= ½×1½ = 1

Yes, including 12 + 14 + 18 + ... etc equals exactly 1.


Don"t think me? just look in ~ this square:

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

we end up through the totality thing!

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Recurring Decimal

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


Example: calculate 0.999...

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

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And now we can use the formula:

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Yes! 0.999... does same 1.

See more: What Instrument Is Used To Measure Atmospheric Pressure ? Atmospheric Pressure


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


Sequences Arithmetic Sequences and also Sums Sigma Notation Algebra Index