Arithmetic Sequences and Sums

Sequence

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

Arithmetic Sequence

In an Arithmetic Sequence the difference between one term and the next is a constant.

In other words, you just add the same value each time ... infinitely.

Example:

1, 4, 7, 10, 13, 16, 19, 22, 25, ...

This sequence has a difference of 3 between each number.

In General you could write an arithmetic sequence like this:

{a, a+d, a+2d, a+3d, ... }

where:

a is the first term, and
d is the difference between the terms (called the "common difference")

Example: (continued)

1, 4, 7, 10, 13, 16, 19, 22, 25, ...

Has:

a = 1 (the first term)
d = 3 (the "common difference" between terms)

And we get:

{a, a+d, a+2d, a+3d, ... }

{1, 1+3, 1+2×3, 1+3×3, ... }

{1, 4, 7, 10, ... }

Rule

You can write an Arithmetic Sequence as a rule:

x_n = a + d(n-1)

(We use "n-1" because d is not used in the 1st term).

Example: Write the Rule, and calculate the 4th term for

3, 8, 13, 18, 23, 28, 33, 38, ...

This sequence has a difference of 5 between each number.

The values of a and d are:

a = 3 (the first term)
d = 5 (the "common difference")

The Rule can be calculated:

x_n = a + d(n-1)

= 3 + 5(n-1)

= 3 + 5n - 5

= 5n - 2

So, the 4th term is:

x₄ = 5×4 - 2 = 18

Is that right? Check for yourself!

Arithmetic Sequences are sometimes called Arithmetic Progressions (A.P.’s)

Summing an Arithmetic Series

To sum up the terms of this arithmetic sequence:

a + (a+d) + (a+2d) + (a+3d) + ...

use this formula:

Sigma

What is that funny symbol? It is called Sigma Notation

(called Sigma) means "sum up"

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

Sigma Notation

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

Here is how to use it:

Example: Add up the first 10 terms of the arithmetic sequence:

{ 1, 4, 7, 10, 13, ... }

The values of a, d and n are:

a = 1 (the first term)
d = 3 (the "common difference" between terms)
n = 10 (how many terms to add up)

So:

Sigma

Becomes:

Sigma

= 5(2+9·3) = 5(29) = 145

Check: why don't you add up the terms yourself, and see if it comes to 145

Why Does the Formula Work?

I want to show you why the formula works, because we get to use an interesting "trick" which is worth knowing.

First, we will call the whole sum "S":

S = a + (a + d) + ... + (a + (n-2)d) + (a + (n-1)d)

Next, rewrite S in reverse order:

S = (a + (n-1)d) + (a + (n-2)d) + ... + (a + d) + a

Now add those two, term by term:

S	=	a	+	(a+d)	+	...	+	(a + (n-2)d)	+	(a + (n-1)d)
S	=	(a + (n-1)d)	+	(a + (n-2)d)	+	...	+	(a + d)	+	a

2S	=	(2a + (n-1)d)	+	(2a + (n-1)d)	+	...	+	(2a + (n-1)d)	+	(2a + (n-1)d)

Each term is the same! And there are "n" of them so ...

2S = n × (2a + (n-1)d)

Now, just divide by 2 and we get:

S = (n/2) × (2a + (n-1)d)

Which is our formula:

Sigma