9-1/9-2: Arithmetic Sequences and SeriesObjectives:•Find the nth term and

arithmetic means of an arithmetic sequence•find the sum of n terms of an arithmetic series

© 2003 Roy L. Gover (roygover@att.net)

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?1,2,4,8,16

1,3,5,7,9

48,40,32,24,16,8,0

Definition

Another 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.

31

43 55

Try ThisFind the next 2 terms in the sequence:

r+15,r+8,r+1,_____,_____r-6 r-13

Definition

The 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?

5th term is 9n-1 term when n=5 is 7

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,...a68=16+(68-1)(-9)=-587a1 n-1 d

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=8

a1=-115

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

arithmetic means

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

Strategy: 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,…31

Series: 1+3+5+…+31

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

1( )2n n

ns a a

Proof...

ExampleFind the sum of the 27 terms in the series:

-14-8-2-…+142

Hint: n=27, a1=-14 & an=142

Try ThisFind the sum of the 13 terms in the series:

-5+1+7+…43

Hint: n=13, a1=-5 & an=43 247

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 don’t know...

StrategyDevelop a new formula for the sum of a series from:

1( )2n n

ns a a an=a1+(n-1)d&

12 ( 1)2n

ns a n d

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

32

322( 12) (32 1)(6)

2s

2

n1a 1n d

Try This

Find 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 Close

What is the difference between a sequence and a series?

Assignment

9.1: pgs. 649-651: 1 – 105 EOO

9.2: pgs. 659-661: 1 – 93 EOO