Geometric Sequences and Sums

Sequence

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

2, 4, 6, 8, …

Geometric Sequences

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

Example

1, 3, 9, 27, 81, 243, …

This sequence has a factor of 3 between each number.

Each term (except the first term) is found by multiplying the previous term by 3.

General Form

In General, we write a Geometric Sequence like this:

{a, ar, ar2, ar3, … }

where:

  • a is the first term
  • r is the factor between the terms (common ratio)
Example

{1,3,9,27,…}

The sequence starts at 1 and triples each time, so

a=1 (the first term)

r=3 (the common ratio)

And we get:

{a, ar, ar2, ar3, … } = {1, 1×3, 1×32, 1×33, … }

{a, ar, ar2, ar3, … } = {1, 3, 9, 27, … }

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)

Example

10, 20, 40, 80, 160, 320, …

This sequence has a factor of 2 between each number.

The values of a and r are:

  • a = 10 (first term)
  • r = 2 (common ratio)

The Rule for any term is:

xn = 10 × 3(n-1)

The 4th term is:

x4 = 10×2(4-1) = 10×23 = 10×8 = 80

And the 10th term is:

x10 = 10×2(10-1) = 10×29 = 10×512= 5120

A Geometric Sequence can also have smaller and smaller values:

Example

8, 4, 2, 1, ½, ¼, …

This sequence has a factor of ½ (a half) between each number.

Its Rule is xn = 4 × (½)n-1

Why “Geometric” Sequence?

Because it is like increasing the dimensions in geometry:

ra line is 1-dimensional and has a length of r
r2in 2 dimensions a square has an area of r2
r3in 3 dimensions a cube has volume r3
 
rnin n dimensions a cube has volume rn

Summing a Geometric Series

To sum a Geometric Sequence, here is a handy formula.

To sum

a + ar + ar2 + … + ar(n-1)

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

Use this formula:

Geometric Sequences and Sums
  • a is the first term
  • r is the “common ratio” between terms
  • n is the number of terms

Just substitute the values of a, r and n

Example

Sum the first 4 terms of

10, 20, 40, 80, 160, 320, …

This sequence has a factor of 2 between each number.

The values of a, r and n are:

  • a = 10 (first term)
  • r = 2 (common ratio)
  • n = 4 (to sum the first 4 terms)

So:

Geometric Sequences and Sums 2

Check it manually:

10 + 20 + 40 + 80 = 150

It is easier to just add them in this example, as there are only 4 terms. But imagine adding 100 terms, using the formula is much easier.

Infinite Geometric Series

So what happens when n goes to infinity?

When r is less than 1, then rn goes to 0 and we get:

Geometric Sequences and Sums 3

NOTE:

  • this does not work when r is 1 or more (or less than -1):
  • r must be between (not including) -1 and 1
  • r should not be 0 because we get the sequence {a,0,0,…} which isn’t geometric
Example

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

{½, ¼, ⅛ … }

We have:

a = ½ (the first term)

r = ½ (halves each time)

And so:

Geometric Sequences and Sums 4

Adding ½ + ¼ + ⅛ + … etc equals exactly 1.

Learn More

Arithmetic Sequences and Series Sums

Trigonometric Identities

Degree of Polynomial

Exponents

Algebra Index

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