1. 9-1/9-2: Arithmetic Sequences and Series
Objectives:
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

2. Example
If 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.

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

4. Definition
An 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

5. Example
Which 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

6. Definition Another definition for arithmetic sequence is:
a1,a1+d, a1+2d,...

7. Example Find 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.

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

9. Try This -5,7,19,___,___,___ Find the next 3 terms in the sequence: 3143 55
Hint: Find d and add it to the prior term to find the next term.

10. Try This r-6 r-13 r+15,r+8,r+1,_____,_____
Find the next 2 terms in the sequence:
r-6
r-13
r+15,r+8,r+1,_____,_____

11. Definition an=a1+(n-1)d =an-1+d
The nth term of an arithmetic sequence is:
an=a1+(n-1)d
=an-1+d

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

13. Example
Find 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

14. Try This a68=16+(68-1)(-9)=-587 a1 n-1 d
Find the 68 term (n=68) of the arithmetic sequence:
16,7,-2,...
a68=16+(68-1)(-9)=-587 a1
n-1 d

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

16. Try This
Find the the first term, a1, in the arithmetic sequence for which a44=229 and d=8
a1=-115

17. Example
Find an arithmetic sequence that has 5 arithmetic means between -11 and 19:
-11,___,___,___,___,___,19
arithmetic means

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

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

20. Definition
An arithmetic series is the sum of the individual terms of an arithmetic sequence.
Sequence: 1,3,5,…31
Series: …+31

21. Definition The sum of the first n terms of an arithmetic series is:
Proof...

22. Example Find the sum of the 27 terms in the series: -14-8-2-…+142
Hint: n=27, a1=-14 & an=142

23. Try This Find the sum of the 13 terms in the series: -5+1+7+…43
Hint: n=13, a1=-5 & an=43
247

24. Example
Find the sum of the first 32 terms in the arithmetic series:
Problem: you know the first term, the number of terms but you don’t know...

25. Strategy
Develop a new formula for the sum of a series from:
&
an=a1+(n-1)d

26. Example
Find the sum of the first 32 terms in the arithmetic series:

27. Try This
Find the sum of the first 32 terms in the arithmetic series:
140

28. Try This
If 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

29. Lesson Close
What is the difference between a sequence and a series?

30. Assignment 9.1: pgs. 649-651: 1 – 105 EOO