Geometric Sequences
In a Geometric Sequence each term is found by multiplying the previous term by a constant.
You are watching: 32+16+8+4+2+1
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")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 geometricThe 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 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). |
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 r is the "common ratio" between terms n is the number of terms
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
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 state ... then the formula is much easier.
Example: Grains of Rice on a Chess Board
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 squto be ...
... 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:
Example: Add 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 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:
S − S·r = a − arn
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:
Example: Add 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:
= 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:
Example: Calculate 0.999...
We can write a recurring decimal as a sum like this:
And now we can use the formula:
Yes! 0.999... does equal 1.
See more: The Shih Tzu Life Span: How Long Can A Shih Tzu Live, What Is The Life Expectancy Of A Shih Tzu Dog
So there we have ins ... Geometric Sequences (and their sums) can do all sorts of amazing and powerful things.
Sequences Arithmetic Sequences and Sums Sigma Notation Algebra Index