Arithmetic Sequence

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

An arithmetic sequence is a finite sequence of numbers with a common difference.^[2] 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

Step	Example	Directions
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..3	Now 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.

Item	Arithmetic Sequence	First Term	Common 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

References

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.

Image Credits

• All images and manipulatives are by David McAdams unless otherwise stated. All images by David McAdams are Copyright © Life is a Story Problem LLC and are licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

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.)