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9-1/9-2: Arithmetic Sequences and Series Objectiv es: Find the nth term and arithmetic means of an arithmetic sequence find the sum of n terms of an © 2003 Roy L. Gover ([email protected])

9-1/9-2: Arithmetic Sequences and Series Objectives: Find the n th term and arithmetic means of an arithmetic sequence find the sum of n terms of an arithmetic

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  • 9-1/9-2: Arithmetic Sequences and SeriesObjectives:Find the nth term and arithmetic means of an arithmetic sequencefind the sum of n terms of an arithmetic series

    2003 Roy L. Gover ([email protected])

  • ExampleIf I gave you 1 dollar today, 2 dollars tomorrow, 3 dollars the next day, etc for 31 days, how many dollars would you have?You will work this problem at the end of the lesson.

  • DefinitionA sequence is a set of numbers in a specific order, for example: 3,6,9,12 or 3,6,10,15,21,...

  • DefinitionAn arithmetic sequence is a sequence in which each term has a common difference. It is represented by: a1, a2, a3,ai, aj, an where the common difference: d=aj-ai

  • ExampleWhich of the following is an arithmetic sequence? What is the common difference?1,2,4,8,161,3,5,7,948,40,32,24,16,8,0

  • DefinitionAnother definition for arithmetic sequence is: a1,a1+d, a1+2d,...

  • ExampleFind the next 3 terms in the sequence:-12,-1,10,___,___,___Hint: Find d and add it to the prior term to find the next term.

  • ExampleFind the next 3 terms in the sequence: r-4,r-1,r+2,___,___,___Hint: Find d and add it to the prior term to find the next term.

  • Try ThisFind the next 3 terms in the sequence: -5,7,19,___,___,___Hint: Find d and add it to the prior term to find the next term.314355

  • Try ThisFind the next 2 terms in the sequence: r+15,r+8,r+1,_____,_____r-6r-13

  • DefinitionThe nth term of an arithmetic sequence is: an=a1+(n-1)d =an-1+d

  • ExampleIn the following arithmetic sequence, name the 5th term:1,3,5,7,9,11,13,...If n=5, the 5th term is 9, what is the n-1 term?

  • ExampleFind the 41st term (n=41) of the arithmetic sequence:11,4,-3,...Steps: 1.Find the value of d 2.Apply the formula: an=a1+(n-1)d

  • Try ThisFind the 68 term (n=68) of the arithmetic sequence:16,7,-2,...

  • ExampleFind the the first term, a1, in the arithmetic sequence for which a31=197 and d=10Hint: use an=a1+(n-1)d with n=31, n-1=30 & an=197

  • Try ThisFind the the first term, a1, in the arithmetic sequence for which a44=229 and d=8a1=-115

  • ExampleFind an arithmetic sequence that has 5 arithmetic means between -11 and 19:-11,___,___,___,___,___,19arithmetic means

  • AnalysisFind an arithmetic sequence that has 5 arithmetic means between -11 and 19:-11,___,___,___,___,___,19Strategy: find d when a1=-11, n=7, & an=19

  • Try ThisFind an arithmetic sequence that has 6 arithmetic means between -12 and 23:-12,__,__,__,__,__,__,23Strategy: find d when a1=-12, n=8, & an=23-12,-7,-2,3,8,13,18,23

  • DefinitionAn arithmetic series is the sum of the individual terms of an arithmetic sequence.Sequence: 1,3,5,31Series: 1+3+5++31

  • DefinitionThe sum of the first n terms of an arithmetic series is:Proof...

  • ExampleFind the sum of the 27 terms in the series: -14-8-2-+142Hint: n=27, a1=-14 & an=142

  • Try ThisFind the sum of the 13 terms in the series: -5+1+7+43Hint: n=13, a1=-5 & an=43247

  • ExampleFind the sum of the first 32 terms in the arithmetic series: -12-6-0-...Problem: you know the first term, the number of terms but you dont know...

  • StrategyDevelop a new formula for the sum of a series from:an=a1+(n-1)d&

  • ExampleFind the sum of the first 32 terms in the arithmetic series: -12-6-0-...

  • Try ThisFind the sum of the first 32 terms in the arithmetic series: 0.5+0.75+1...140

  • Try ThisIf I gave you 1 dollar today, 2 dollars tomorrow, 3 dollars the next day, etc for 31 days, how many dollars would you have?$496

  • Lesson CloseWhat is the difference between a sequence and a series?

  • Assignment9.1: pgs. 649-651: 1 105 EOO 9.2: pgs. 659-661: 1 93 EOO