1. 11.2 Arithmetic Sequences & Series
p.659
What is an arithmetic sequence?
What is the rule for an arithmetic sequence?
How do you find the rule when given two terms?

2. Arithmetic Sequence:
The difference between consecutive terms is constant (or the same).
The constant difference is also known as the common difference (d).

3. Example: Decide whether each sequence is arithmetic.
5,11,17,23,29,…
11-5=6
17-11=6
23-17=6
29-23=6
Arithmetic (common difference is 6)
-10,-6,-2,0,2,6,10,…
-6--10=4
-2--6=4
0--2=2
2-0=2
6-2=4
10-6=4
Not arithmetic (because the differences are not the same)

4. Rule for an Arithmetic Sequence
an=a1+(n-1)d

5. Example: Write a rule for the nth term of the sequence 32,47,62,77,…
Example: Write a rule for the nth term of the sequence 32,47,62,77,… . Then, find a12.
There is a common difference where d=15, therefore the sequence is arithmetic.
Use an=a1+(n-1)d
an=32+(n-1)(15)
an=32+15n-15
an=17+15n
a12=17+15(12)=197

6. Example: One term of an arithmetic sequence is a8=50
Example: One term of an arithmetic sequence is a8=50. The common difference is Write a rule for the nth term.
Use an=a1+(n-1)d to find the 1st term!
a8=a1+(8-1)(.25)
50=a1+(7)(.25)
50=a1+1.75
48.25=a1
* Now, use an=a1+(n-1)d to find the rule.
an=48.25+(n-1)(.25)
an= n-.25
an=48+.25n

7. Now graph an=48+.25n.
Just like yesterday, remember to graph the ordered pairs of the form (n,an)
So, graph the points (1,48.25), (2,48.5), (3,48.75), (4,49), etc.

8. an=a1+(n-1)d an=-6+(n-1)(4) OR an=-10+4n
Example: Two terms of an arithmetic sequence are a5=10 and a30=110. Write a rule for the nth term.
Begin by writing 2 equations; one for each term given.
a5=a1+(5-1)d OR 10=a1+4d
And
a30=a1+(30-1)d OR 110=a1+29d
Now use the 2 equations to solve for a1 & d.
10=a1+4d
110=a1+29d (subtract the equations to cancel a1)
-100= -25d
So, d=4 and a1=-6 (now find the rule)
an=a1+(n-1)d
an=-6+(n-1)(4) OR an=-10+4n

9. Example (part 2): using the rule an=-10+4n, write the value of n for which an=-2.

10. What is an arithmetic sequence?
The difference between consecutive terms is a constant
What is the rule for an arithmetic sequence?
an=a1+(n-1)d
How do you find the rule when given two terms?
Write two equations with two unknowns and use linear combination to solve for the variables.

11. Assignment
p. 663, odd

12. Arithmetic Sequences and Series day 3
What is the formula for find the sum of a finite arithmetic series?

13. Arithmetic Series The sum of the terms in an arithmetic sequence
The formula to find the sum of a finite arithmetic series is:
Last Term
1st Term
# of terms

14. Example: Consider the arithmetic series 20+18+16+14+… .
Find the sum of the 1st 25 terms.
First find the rule for the nth term.
an=22-2n
So, a25 = -28 (last term)
Find n such that Sn=-760

15. Always choose the positive solution!
-1520=n( n)
-1520=-2n2+42n
2n2-42n-1520=0
n2-21n-760=0
(n-40)(n+19)=0
n=40 or n=-19
Always choose the positive solution!

16. What is the formula for find the sum of a finite arithmetic series?

17. Assignment:
p. 663
45-56 all
Skip 49 & 50