1. Chapter 7: Sequences & Series (pgs. 434-479)
Katia, Lauren, Bridget, Filip

2. Extremely Important Equations
Arithmetic
Geometric
Rule for nth Term
an= d(n-1) + a1
an= a1(r)n-1
Finite Sum (terminates)
Sn= n((a1+an) ⁄2)
Sn= a1 ((1-rn) ⁄ (1-r))
Infinite Sum X
S∞= a1 ⁄ (1-r)

3. Identifying Seq. & Series
Neither
No common ratio/difference between each term in a pattern
2,3,6,18,108,1944
1944/108=18
108/18=6
Not geometric
=1836
108-18=90
Not arithmetic so neither
Arithmetic
Pattern of numbers that increase or decrease by adding a common ratio
1,4,7,10,13,16
16-13=3
13-10=3
10-7=3
7-4=3
4-1=3
All have the same difference
Geometric
Pattern of numbers that increase/decrease by multiplying a common ratio.
1,2,4,8,16,36
36/16=2
16/8=2
8/4=2
4/2=2
2/1=2
All have the same quotient

4. Writing Recursive Rules

5. The Three R.R’s Special Sequences
Some sequences are neither arithmetic nor geometric, and therefore require their own “type” of recursive rule.
Ex: 1, 2, 3, 5, 8, 13
The previous two terms, an-1 and an-2, add together to make the current term, an
Rule: an= an-2 + an-1
Arithmetic
an= an-1 + d (d= common difference)
Ex: 1, 4, 7, 10, 13, 16
d= 3
Rule: an= an-1 + 3
Geometric
an= an-1 r (r= common ratio)
Ex: 4, 32, 256, 2048, 16384
r= 8
Rule: an= an-1 8

6. Writing Arithmetic and Geometric Rules

7. Writing Arithmetic and Geometric Rules
an= d(n-1) + a1
d= common difference
n= # in sequence
a1= 1st term
Example:
5, 11, 17, 23, 29, ...
an=6(n-1)+5
an=6n-6+5
an=6n-1
an= a1(r)n-1
r=common ratio
6, 18, 54, 162, …
an=6(3)n-1 +6 ^ +6 ^ X3 ^ X3 ^

8. Practice Writing Equations
3, -3, -9, -15, -21, … a2=8 a4= (geometric)

9. Sums of series

10. Sum of a finite arithmetic series:
Sn=n(a1+an/2)
Sn is the mean of the first and nth terms, multiplied by the number of terms
Example: Find the sum of 2, 6, 10, 14,

11. Sum of a finite geometric series:
Sn=a1(1-rn/1-r)
Sn refers to the sum of the series
A1 refers to the first number in the series
r is the rule
n is the number of terms

12. Sum of an infinite geometric series:
Sn=a1/(1-r)

13. Word Problemos

14. Sample Word Problem
Jimmy is building a tower out of wooden blocks. The bottom first row has 33 blocks, and each following row has 3 less blocks in it. How many blocks are in the 5th row?

15. Step 1: Determine what sequence this is… arithmetic, geometric, or neither.
A: arithmetic
Step 2: Write a rule to find the nth term for this problem.
A: an= -3n + 36
Step 3: Substitute 5 for n
A: a5= -3(5) + 36
Step 4: Simplify
A: a5= Blocks