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, we 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 we 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

We 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:

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:

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:

Becomes:

= 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?

Let's see 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:

Indices & the Law of Indices

Introduction

Indices are a useful way of more simply expressing large numbers. They also present us with many useful properties for manipulating them using what are called the Law of Indices.

What are Indices?

The expression 2⁵ is defined as follows:

We call "2" the base and "5" the index.

Law of Indices

To manipulate expressions, we can consider using the Law of Indices. These laws only apply to expressions with the same base, for example, 3⁴ and 3² can be manipulated using the Law of Indices, but we cannot use the Law of Indices to manipulate the expressions 3⁵ and 5⁷ as their base differs (their bases are 3 and 5, respectively).