Evaluate The Following Geometric Sum

Geometric Sequences and Sums

Sequence

A Sequence is a set of things (usually numbers) that are in order.

Sequence

Geometric Sequences

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

Instance:

1, 2, 4, 8, 16, 32, 64, 128, 256 , ...

This sequence has a gene of ii between each number.

Each term (except the first term) is found past multiplying the previous term past ii.

geometric sequence 1,2,4,8,16,

In General we write a Geometric Sequence like this:

{a, ar, ar², ar³, ... }

where:

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

Example: {i,ii,4,8,...}

The sequence starts at 1 and doubles each fourth dimension, so

a=one (the outset term)
r=2 (the "common ratio" between terms is a doubling)

And we get:

{a, ar, ar², ar³, ... }

= {1, i×2, ane×two², one×twoⁱⁱⁱ, ... }

= {1, 2, 4, 8, ... }

But exist conscientious, r should not be 0:

When r=0, we get the sequence {a,0,0,...} which is not geometric

The Rule

We can besides calculate any term using the Rule:

x_n = ar^(n-ane)

(We use "n-one" because ar⁰ is for the 1st term)

Example:

x, 30, ninety, 270, 810, 2430, ...

This sequence has a gene of 3 between each number.

The values of a and r are:

a = x (the offset term)
r = 3 (the "common ratio")

The Rule for any term is:

x_{due north} = x × 3^(north-1)

So, the 4th term is:

x₄ = ten×3^(iv-ane) = 10×3³ = 10×27 = 270

And the tenth term is:

x_x= 10×iii^(x-one) = ten×3⁹ = 10×19683 = 196830

A Geometric Sequence tin can also have smaller and smaller values:

Instance:

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

Its Dominion is x_n = 4 × (0.five)^n-1

Why "Geometric" Sequence?

Because information technology is like increasing the dimensions in geometry:

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

Geometric Sequences are sometimes called Geometric Progressions (G.P.'southward)

Summing a Geometric Series

To sum these:

a + ar + ar² + ... + ar^(north-1)

(Each term is ar^g , where k starts at 0 and goes up to due north-1)

Nosotros can use this handy formula:

Sigma

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

(chosen Sigma) ways "sum up"

And below and higher up it are shown the starting and ending values:

Sigma Notation

It says "Sum up due north where north goes from ane to 4. Answer=ten

The formula is easy to apply ... only "plug in" the values of a, r and north

Instance: Sum the kickoff four terms of

10, 30, 90, 270, 810, 2430, ...

This sequence has a gene of 3 between each number.

The values of a, r and n are:

a = 10 (the first term)
r = three (the "common ratio")
n = iv (we desire to sum the offset four terms)

So:

Sigma

Becomes:

Sigma

You can check it yourself:

ten + thirty + xc + 270 = 400

And, aye, it is easier to just add them in this instance, equally there are but 4 terms. Only imagine adding fifty terms ... so the formula is much easier.

Using the Formula

Let's see the formula in action:

Example: Grains of Rice on a Chess Board

chess board

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

When nosotros identify rice on a chess board:

1 grain on the outset foursquare,
2 grains on the 2d square,
4 grains on the third then on,
...

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

... how many grains of rice in full?

And then we have:

a = ane (the first term)
r = ii (doubles each fourth dimension)
northward = 64 (64 squares on a chess board)

And so:

Sigma

Becomes:

Sigma

= ane−ii⁶⁴ −1 = ii⁶⁴ − 1

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

Which was exactly the result nosotros got on the Binary Digits folio (thank goodness!)

And another example, this fourth dimension with r less than 1:

Instance: Add together upward the first 10 terms of the Geometric Sequence that halves each time:

{ 1/2, 1/4, 1/viii, 1/16, ... }

The values of a, r and due north are:

a = ½ (the first term)
r = ½ (halves each time)
northward = 10 (ten terms to add together)

So:

Sigma

Becomes:

Sigma

Very close to 1.

(Question: if nosotros go on to increase n, what happens?)

Why Does the Formula Piece of work?

Permit'southward see why the formula works, because we go to use an interesting "play a trick on" which is worth knowing.

First, telephone call the whole sum "S": South = a + ar + arⁱⁱ + ... + ar⁽ⁿ⁻ⁱⁱ⁾ + ar⁽ⁿ⁻¹⁾

Next, multiply S by r: Southward·r = ar + ar² + ar³ + ... + ar^(n−one) + arⁿ

Notice that S and South·r are like?

Now subtract them!

Proof

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

By subtracting Southward·r from Southward nosotros get a simple result:

S − Due south·r = a − arⁿ

Let'southward rearrange it to observe S:

Factor out S and a: S(1−r) = a(i−r^northward)

Divide by (1−r): S = a(ane−rⁿ) (ane−r)

Which is our formula (ta-da!):

Sigma

Infinite Geometric Series

And so what happens when due north goes to infinity?

We tin apply this formula:

Sigma

But be conscientious:

r must be betwixt (but not including) −1 and one

and r should not be 0 because the sequence {a,0,0,...} is non geometric

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

Let'southward bring back our previous example, and see what happens:

Example: Add up ALL the terms of the Geometric Sequence that halves each time:

{ 1 2 , 1 four , 1 8 , 1 xvi , ... }

We have:

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

And so:

Sigma

= ½×1 ½ = ane

Yes, adding 1 ii + one 4 + 1 eight + ... etc equals exactly 1.

Don't believe me? Just look at this foursquare:

By adding up ane ii + 1 four + 1 viii + ...

we finish up with the whole thing!

Recurring Decimal

On another page we asked "Does 0.999... equal ane?", well, allow us see if we can calculate information technology:

Case: Summate 0.999...

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

Sigma

And at present we tin can use the formula:

Sigma

Yes! 0.999... does equal 1.

So there nosotros accept it ... Geometric Sequences (and their sums) can practise all sorts of amazing and powerful things.