# Arithmetic Sequence

Pronunciation: /æ.rɪθˈmɛ.tɪk ˈsi.kwəns/ Explain

An arithmetic sequence is a finite sequence of numbers with a common difference. For example, the arithmetic sequence { 1, 3, 5, 7 } has a common difference of 2 since 1 + 2 = 3, 3 + 2 = 5 and 5 + 2 = 7.

### Notation for Arithmetic Sequence

Arithmetic sequence can be denoted {a0 + k · d}k=0..n-1 where:

• a0 is the value of the first term;
• d is the common difference; and
• n is the number of terms in the sequence.

Take the arithmetic sequence { 1, 3, 5, 7 }. Since the first term is 1, a0 = 1. The common difference is 3 - 1 = 2, so d = 2. There are 4 terms in the sequence, so n = 4. This gives the expression { 1 + 2k }k=0..3.

### Discovery

StepExampleDirections
1 Take a piece of paper.
2 Using a ruler, make tick marks along opposite edges.
3 Using a straightedge, connect the tick marks.
4 Using scissors, cut off one of the strips. How many strips do you have so far? Write it like this { 1 }.
5 Cut off another strip of paper. How many strips do you have now? Write it like this { 1, 2 }.
6 Continue cutting off strips until the paper is completely cut. Write down the sequence.
7(1 + k)k=0..3Now write the sequence using arithmetic sequence notation.

#### Understanding Check

Write down your answer to the following questions. Then click on the 'answer' icon to see if you understand.

ItemArithmetic
Sequence
First TermCommon
Difference
Number of
Terms
Expression
1{ 4, 7, 10, 13, 16 }
Click for Answer
4
Click for Answer
3
Click for Answer
5
Click for Answer
{4 + 3·k}k=0..4
2{ -1, 1, 3, 5, 7, 9 }
Click for Answer
-1
Click for Answer
2
Click for Answer
6
Click for Answer
{-1 + 2·k}k=0..5
3
Click for Answer
{ 6, 8, 10, 12 }
Click for Answer
6
Click for Answer
2
Click for Answer
4
{ 6 + 2·k }k=0..3
4
Click for Answer
{ 9, 6, 3, 0, -3 }
Click for Answer
9
Click for Answer
-3
Click for Answer
5
{ 9 - 3·k }k=0..4
1. McAdams, David E.. All Math Words Dictionary, arithmetic sequence. 2nd Classroom edition 20150108-4799968. pg 20. Life is a Story Problem LLC. January 8, 2015. Buy the book
2. arithmetic progression. merriam-webster.com. Encyclopedia Britannica. Merriam-Webster. Last Accessed 6/12/2018. https://www.merriam-webster.com/dictionary/arithmetic%20progression. Buy the book
3. Bettinger, Alvin K. and Englund, John A.. Algebra and Trigonometry. pp 260-262. www.archive.org. International Textbook Company. January 1963. Last Accessed 6/12/2018. http://www.archive.org/stream/algebraandtrigon033520mbp#page/n277/mode/1up/search/progression. Buy the book
4. Jones, Burton. Elementary Concepts of Mathematics. pp 118-122. www.archive.org. MacMillan and Company. 1947. Last Accessed 6/12/2018. http://www.archive.org/stream/elementaryconcep029487mbp#page/n138/mode/1up. Buy the book

### Cite this article as:

McAdams, David E. Arithmetic Sequence. 4/12/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/a/arithmeticsequence.html.

### Revision History

4/12/2019: Changed equations and expressions to new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
6/15/2018: Removed broken links, updated license, implemented new markup. (McAdams, David E.)
1/6/2010: Added "References". (McAdams, David E.)
4/5/2008: Adding discovery activity. (McAdams, David E.)
2/3/2008: Added color to example of common difference. (McAdams, David E.)
8/7/2007: Initial version. (McAdams, David E.)

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