# Intro to arithmetic sequences

CCSS Math: HSF.IF.A.3
Get comfortable with sequences in general, and learn what arithmetic sequences are.
Before you take this lesson, make sure you know how to add and subtract negative numbers.

# What is a sequence?

Here are a few lists of numbers:
• 3, 5, 7 ...
• 21, 16, 11, 6 ...
• 1, 2, 4, 8 ...
Ordered lists of numbers like these are called sequences. Each number in a sequence is called a term.
$3,$$5,$$7,...$
$\uparrow$$\uparrow$$\uparrow$
$\footnotesize 1^\text{st}\text{ term}$$\footnotesize 2^\text{nd}\text{ term}$$\footnotesize 3^\text{rd}\text{ term}$

Sequences usually have patterns that allow us to predict what the next term might be.
For example, in the sequence 3, 5, 7 ..., you always add two to get the next term:
$\footnotesize\maroonC{+2\,\Large\curvearrowright}$$\footnotesize\maroonC{+2\,\Large\curvearrowright}$
$3,$$5,$$7,...$

The three dots that come at the end indicate that the sequence can be extended, even though we only see a few terms.
We can do so by using the pattern.
For example, the fourth term of the sequence should be nine, the fifth term should be 11, etc.
$\footnotesize\maroonC{+2\,\Large\curvearrowright}$$\footnotesize\maroonC{+2\,\Large\curvearrowright}$$\footnotesize\maroonC{+2\,\Large\curvearrowright}$$\footnotesize\maroonC{+2\,\Large\curvearrowright}$
$3,$$5,$$7,$$9,$$11,...$

Extend the sequences according to their pattern.

# What is an arithmetic sequence?

For many of the examples above, the pattern involves adding or subtracting a number to each term to get the next term. Sequences with such patterns are called arithmetic sequences.
In an arithmetic sequence, the difference between consecutive terms is always the same.
For example, the sequence 3, 5, 7, 9 ... is arithmetic because the difference between consecutive terms is always two.
$\footnotesize\maroonC{+2\,\Large\curvearrowright}$$\footnotesize\maroonC{+2\,\Large\curvearrowright}$$\footnotesize\maroonC{+2\,\Large\curvearrowright}$
$3,$$5,$$7,$$9,...$

The sequence 21, 16, 11, 6 ... is arithmetic as well because the difference between consecutive terms is always five.
$\footnotesize\maroonC{-5\,\Large\curvearrowright}$$\footnotesize\maroonC{-5\,\Large\curvearrowright}$$\footnotesize\maroonC{-5\,\Large\curvearrowright}$
$21,$$16,$$11,$$6,...$

The sequence 1, 2, 4, 8 ... is not arithmetic because the difference between consecutive terms is not the same.
$\footnotesize\maroonC{+1\,\Large\curvearrowright}$$\footnotesize\maroonC{+2\,\Large\curvearrowright}$$\footnotesize\maroonC{+4\,\Large\curvearrowright}$
$1,$$2,$$4,$$8,...$

# The common difference

The common difference of an arithmetic sequence is the constant difference between consecutive terms.
For example, the common difference of 10, 21, 32, 43 ... is 11:
$\footnotesize\maroonC{+11\,\Large\curvearrowright}$$\footnotesize\maroonC{+11\,\Large\curvearrowright}$$\footnotesize\maroonC{+11\,\Large\curvearrowright}$
$10,$$21,$$32,$$43,...$

The common difference of –2, –5, –8, –11 ... is negative three:
$\footnotesize\maroonC{-3\,\Large\curvearrowright}$$\footnotesize\maroonC{-3\,\Large\curvearrowright}$$\footnotesize\maroonC{-3\,\Large\curvearrowright}$
$-2,$$-5,$$-8,$$-11,...$