C#

     

Geometric Sequences

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

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This sequence has a factor of 2 between each number.

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

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In General we write a Geometric Sequence lượt thích this:

a, ar, ar2, ar3, ...

where:

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


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

The sequence 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 be careful, r should not be 0:

When r=0, we get the sequence a,0,0,... Which is not geometric

The Rule

We can also calculate any term using the Rule:


This sequence has a factor of 3 between each number.

The values of a and r are:

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

The Rule 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 term is:

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


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

Its Rule is xn = 4 × (0.5)n-1


Why "Geometric" Sequence?

Because it is lượt thích increasing the dimensions in geometry:

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


Summing a Geometric Series

To sum these:

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

(Each term is ark, where k starts at 0 và goes up lớn 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 called Sigma Notation

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(called Sigma) means "sum up"

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

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It says "Sum up n where n goes from 1 lớn 4. Answer=10


This sequence has a factor of 3 between each number.

The values of a, r & 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 check it yourself:

10 + 30 + 90 + 270 = 400

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


Example: Grains of Rice on a Chess Board

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

When we place rice on a chess board:

1 grain on the first square, 2 grains on the second square, 4 grains on the third & so on, ...

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

... How many grains of rice in total?

So we have:

a = 1 (the first term) r = 2 (doubles each time) n = 64 (64 squares on 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 got on the Binary Digits page (thank goodness!)


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


Example: địa chỉ up the first 10 terms of the Geometric Sequence that halves each time:

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

The values of a, r & n are:

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

So:

Becomes:

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Very close lớn 1.

(Question: if we continue khổng lồ increase n, what happens?)


Why Does the Formula Work?

Let"s see why the formula works, because we get lớn use an interesting "trick" which is worth knowing.


First, điện thoại tư vấn the whole 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 & S·r are similar?

Now subtract them!

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

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


S − S·r = a − arn


Let"s rearrange it khổng lồ find S:


Factor out S
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 can use this formula:

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But be careful:


r must be between (but not including) −1 and 1

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


So our infnite geometric series has a finite sum when the ratio is less than 1 (and greater than −1)

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


Example: địa chỉ up ALL the terms of the Geometric Sequence that 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, adding 12 + 14 + 18 + ... etc equals exactly 1.


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

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

we over up with the whole thing!

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Recurring Decimal

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


Example: Calculate 0.999...

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

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And now we can use the formula:

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Yes! 0.999... does equal 1.

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So there we have it ... Geometric Sequences (and their sums) can do all sorts of amazing & powerful things.


Sequences Arithmetic Sequences & Sums Sigma Notation Algebra Index