Arithmetic Sequence

Pronunciation: /əˈrɪθmə tɪk ˈsik wəns/ Explain

An arithmetic sequence is a finite sequence of numbers with a common difference.[1] 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
1Blank piece of paperTake a piece of paper.
2Piece of paper with tick marks.Using a ruler, make tick marks along opposite edges.
3Piece of paper with lines.Using a straightedge, connect the tick marks.
4Piece of paper with one strip cut off.Using scissors, cut off one of the strips. How many strips do you have so far? Write it like this { 1 }.
5Piece of paper with two strips cut off.Cut off another strip of paper. How many strips do you have now? Write it like this { 1, 2 }.
6Piece of paper with three strips cut off.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

References

  1. arithmetic progression. merriam-webster.com. Encyclopedia Britannica. Merriam-Webster. Last Accessed 8/6/2018. https://www.merriam-webster.com/dictionary/arithmetic%20progression.
  2. Bettinger, Alvin K. and Englund, John A.. Algebra and Trigonometry. pp 260-262. www.archive.org. International Textbook Company. January 1963. Last Accessed 8/6/2018. http://www.archive.org/stream/algebraandtrigon033520mbp#page/n277/mode/1up/search/progression. Buy the book
  3. Jones, Burton. Elementary Concepts of Mathematics. pp 118-122. www.archive.org. MacMillan and Company. 1947. Last Accessed 8/6/2018. http://www.archive.org/stream/elementaryconcep029487mbp#page/n138/mode/1up. Buy the book

Cite this article as:

McAdams, David E. Arithmetic Sequence. 6/15/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/a/arithmeticsequence.html.

Image Credits

Revision History

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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