What is the 3rd term of a geometric sequence if its first term is 6 and a common ratio of 1 2

An geometric sequence or progression, is a sequence where each term is calculated by multiplying the previous term by a fixed number.

Geometric Sequences

This fixed number is called the common ratio, r.
The common ratio can be positive or negative, an integer or a fraction.

The common ratio can be calculated by dividing any term by the one before it. 
e.g. Common ratio = t n+1 ÷ t n

The first term of a geometric sequence is shown by the variable a.

General Term, tn

A geometric sequence can be written:

Geometric Series

If terms of a geometric sequence are added together a geometric series is formed.

2 + 4 + 8 + 16 is a finite geometric series
2 + 4 + 8 + 16 + ... is an infinte geometric series

To find the sum of the first n terms of a geometric sequence use the formula:

Sum of first n terms of a geometric sequence wherer = common ratioa = first term n = number of terms

OR

If the common ratio is a fraction i.e. -1 < r < 1 then an equivalent formula, shown below is easier to use.

Sum of first n terms of a geometric sequence for when -1 < r < 1 i.e. r is a fraction

Example What is the sum of the first 10 terms of the geometric sequence:       3, 6, 12, ...

Common ratio r = 6 ÷ 3 = 2 Number of terms n = 10 First term a = 3

The Sum to Infinity of a Geometric Sequence

Spreadsheets are very useful for generating sequences and series.

For a geometric sequence with a common ratio greater than 1:

The formula in cell B3 is = B2*2 The formula in cell D3 is =D2 + B3 The fill down command is then used to complete the sequences.

It can be seen that as successive terms are added the sum of the terms increases.
If there were an infinite number of terms the sum would be infinity.

For a geometric sequence with a common ratio less than 1:

The formula in cell B3 is = B2*2 The formula in cell D3 is =D2 + B3 The fill down command is then used to comlete the sequences.

It can be seen that as successive terms are added the sum of the terms appears to be heading towards 16. 
If there were an infinite number of terms the sum would be 16.

This is called the sum to infinity of a geometric sequence and only applies when the common ratio is a fraction

i.e. -1 < r < +1. 
r can be positive or negative.

The following formula can be used:

Sum to infinity of geometric sequence wherer = common ratio a = first term

Example

Find the sum to infinity of the geometric sequence 8, 4, 2, 1, ...

a = 8 and r = 0.5

As can be seen from cell D10 in the spreadsheet above, 16 is the value the sums were heading towards.

To see this concept clearly illustrated - 

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

Geometric Sequences

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

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.

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")

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:

xn = ar(n-1)

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

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

A Geometric Sequence can also have smaller and smaller values:

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 like increasing the dimensions in geometry:

	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 and more dimensions in mathematics).

Geometric Sequences are sometimes called Geometric Progressions (G.P.’s)

Summing a Geometric Series

To sum these:

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

(Each term is ark, where k starts at 0 and goes up to n-1)

We can use this handy formula:

a is the first term

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

(called Sigma) means "sum up"

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

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

The formula is easy to use ... just "plug in" the values of a, r and n

This sequence has a factor of 3 between each number.

The values of a, r and n are:

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

So:

Becomes:

You can check it yourself:

10 + 30 + 90 + 270 = 400

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

Using the Formula

Let's see the formula in action:

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 and so on,
• ...

... 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:

= 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:

The values of a, r and n are:

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

So:

Becomes:

Very close to 1.

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

Why Does the Formula Work?

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

First, call 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 and S·r are similar?

Now subtract them!

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:

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 when n goes to infinity?

We can use this formula:

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:

We have:

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

And so:

= 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 end up with the whole thing!

Recurring Decimal

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

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

And now we can use the formula:

Yes! 0.999... does equal 1.

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

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