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# Intro to arithmetic sequences

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, comma5, comma7, comma, point, point, point
\uparrow\uparrow\uparrow
1, start superscript, start text, s, t, end text, end superscript, start text, space, t, e, r, m, end text2, start superscript, start text, n, d, end text, end superscript, start text, space, t, e, r, m, end text3, start superscript, start text, r, d, end text, end superscript, start text, space, t, e, r, m, end text
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:
start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6
3, comma5, comma7, comma, point, point, point
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.
start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6
3, comma5, comma7, comma9, comma11, comma, point, point, point

Extend the sequences according to their pattern.
Problem 1
Pattern: Add five to the previous term.
start color #ed5fa6, plus, 5, \curvearrowright, end color #ed5fa6start color #ed5fa6, plus, 5, \curvearrowright, end color #ed5fa6start color #ed5fa6, plus, 5, \curvearrowright, end color #ed5fa6
3, comma8, comma13, comma
comma, point, point, point

Problem 2
Pattern: Subtract three from the previous term.
start color #ed5fa6, minus, 3, \curvearrowright, end color #ed5fa6start color #ed5fa6, minus, 3, \curvearrowright, end color #ed5fa6start color #ed5fa6, minus, 3, \curvearrowright, end color #ed5fa6
20, comma17, comma14, comma
comma, point, point, point

Problem 3
Pattern: Multiply the previous term by two.
start color #ed5fa6, times, 2, \curvearrowright, end color #ed5fa6start color #ed5fa6, times, 2, \curvearrowright, end color #ed5fa6start color #ed5fa6, times, 2, \curvearrowright, end color #ed5fa6
3, comma6, comma12, comma
comma, point, point, point

Problem 4
Match each sequence with its 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.
start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6
3, comma5, comma7, comma9, comma, point, point, point
The sequence 21, 16, 11, 6 ... is arithmetic as well because the difference between consecutive terms is always minus five.
start color #ed5fa6, minus, 5, \curvearrowright, end color #ed5fa6start color #ed5fa6, minus, 5, \curvearrowright, end color #ed5fa6start color #ed5fa6, minus, 5, \curvearrowright, end color #ed5fa6
21, comma16, comma11, comma6, comma, point, point, point
The sequence 1, 2, 4, 8 ... is not arithmetic because the difference between consecutive terms is not the same.
start color #ed5fa6, plus, 1, \curvearrowright, end color #ed5fa6start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6start color #ed5fa6, plus, 4, \curvearrowright, end color #ed5fa6
1, comma2, comma4, comma8, comma, point, point, point

Problem 5
Select all arithmetic sequences.

Problem 6
The first term of a sequence is one. Which of the following patterns would make the sequence arithmetic?

## 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:
start color #ed5fa6, plus, 11, \curvearrowright, end color #ed5fa6start color #ed5fa6, plus, 11, \curvearrowright, end color #ed5fa6start color #ed5fa6, plus, 11, \curvearrowright, end color #ed5fa6
10, comma21, comma32, comma43, comma, point, point, point
The common difference of –2, –5, –8, –11 ... is negative three:
start color #ed5fa6, minus, 3, \curvearrowright, end color #ed5fa6start color #ed5fa6, minus, 3, \curvearrowright, end color #ed5fa6start color #ed5fa6, minus, 3, \curvearrowright, end color #ed5fa6
minus, 2, commaminus, 5, commaminus, 8, commaminus, 11, comma, point, point, point

Problem 7
What is the common difference of 2, 8, 14, 20 ...?

Problem 8
What is the common difference of 5, comma, 2, comma, minus, 1, comma, minus, 4, point, point, point?

Problem 9
What is the common difference of 1, comma, 1, start fraction, 1, divided by, 3, end fraction, comma, 1, start fraction, 2, divided by, 3, end fraction, comma, 2, comma, point, point, point?

Reflection question
What must be true about an arithmetic sequence whose common difference is negative?

Challenge problem
The first term of an arithmetic sequence is 10 and its common difference is negative seven.
What is the fourth term of the sequence?

## What's next?

Learn about formulas of arithmetic sequences, which give us the information we need to find any term in the sequence.

## Want to join the conversation?

• is the lucas series series also an arithmetic sequence
eg. {1,1,2,3,5,8,13,21,34...} where asub(k)=asub(k-1)+asub(k-2)
• NO. Take a look at the difference between the terms of the sequence. The difference between the terms is not constant (not the same), hence not an arithmetic sequence.
• So if adding and subtracting from the previous terms create an arithmetic sequence, would multiplying or dividing make a geometric sequence?
• In short, yes.

Arithmetic is always adding or subtracting the same constant term or amount.
Geometric is always multiplying or dividing by the same constant amount.
• Are arithmetic sequences always either addition or subtraction
• Yes that is what makes them arithmetic. Multiply and divide are geometric sequences.
• do all arithmetic sequences have to have real numbers?
• A sequence can be of unreal numbers I think that arithmetic progression should of real numbers
• Instead of learning it in the book, my teacher says to learn on here but its hard when I'm a visual learner XD
• If I multiple the last number by a fixed number in a sequence,is it not that an arithmetic sequences
• Simply put if its multiplied or divided it'll be geometric whereas if its added or subtracted its an arithmetic secuence
(1 vote)
• Is there many kinds of different formulas to write explicit and Recursive equations or there just one?
• There is basically one formula, you just have to change the numbers.
• i understand it but i was wondering if there is an easier
way to solve arithemic sequences
• no i think that this would be the easiest way
(1 vote)
• Can anyone here explain in detail , what is the meaning of Recursive ?
• Use any dictionary website to get the formal definition.

With the recursive equation for a sequence, you must know the value of the prior term to create the next term. So, you follow a repetitive sequence of steps to get to the value you want. For example, to find the 4th term of a sequence using a recursive equation, you:
1) Calculate the 1st term (this is often given to you).
2) Use the value of the 1st term to calculate the 2nd term.
3) Use the value of the 2nd term to calculate the 3rd term.
4) Use the value of the 3rd term to calculate the 4th term.

Basically, you can't get to the 4th term in one step. You have to go term to term up to the 4th term (or what ever term you want). If you want to get to the term directly, the explicit equation for a sequence would do that.