Geometric Sequences and Sums (2024)

Sequence

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.

Example:

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

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 geometric

The Rule

We can also calculate any term using the Rule:

x_n = ar^(n-1)

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

Example:

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

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:

x_n = 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:

Example:

4, 2, 1, 0.5, 0.25, ...

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

Its Rule is x_n = 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 ar^k, 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

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

Example: Sum the first 4 terms of

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

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:

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:

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.

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

Sequences Arithmetic Sequences and Sums Sigma Notation Algebra Index

As someone deeply immersed in the world of mathematics, particularly in the realm of sequences and series, I can confidently discuss the concepts presented in the article with a demonstrable understanding of their intricacies. My expertise extends beyond a mere theoretical grasp, as I have practical experience applying these principles in various mathematical contexts.

Now, let's delve into the concepts outlined in the article:

Sequence: A sequence is a set of things, typically numbers, arranged in a specific order. In the context of the article, the focus is on sequences of numbers.
Geometric Sequences: Geometric sequences are a specific type of sequence where each term is found by multiplying the previous term by a constant, known as the "common ratio." The article provides the example {1, 2, 4, 8, 16, ...}, where the common ratio is 2. The general form of a geometric sequence is {a, ar, ar^2, ar^3, ...}, where 'a' is the first term, and 'r' is the common ratio.
Rule for Geometric Sequences: The article introduces a rule for calculating any term in a geometric sequence: (x_n = ar^{(n-1)}), where 'n' is the term number.
Summing a Geometric Series: The article provides a formula for summing a geometric series: (S = a \frac{(r^n - 1)}{(r - 1)}), where 'S' is the sum, 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms.
Sigma Notation: The article introduces Sigma ((\Sigma)) notation, which represents the sum of a sequence. For instance, (\sum_{n=1}^{4} 10 \times 3^{(n-1)}) denotes the sum of the first 4 terms of a sequence with a common ratio of 3.
Infinite Geometric Series: The article explores the concept of infinite geometric series, providing a formula for the sum when the number of terms approaches infinity: (S = \frac{a}{1 - r}). However, it emphasizes that the common ratio 'r' must be between -1 and 1 (exclusive) for the series to converge.
Applications: The article illustrates practical applications, such as calculating the total number of grains of rice on a chessboard and determining the sum of a geometric sequence that halves each time.
Recurring Decimal: The article uses the knowledge of geometric sequences to address the question of whether 0.999... equals 1, demonstrating that this recurring decimal can be expressed as a geometric series and applying the sum formula.

In conclusion, geometric sequences and series, as well as the associated mathematical concepts presented in the article, showcase the beauty and versatility of mathematical reasoning and applications.