1. Sequences and Series IB standard
Students should know Arithmetic sequence and series; sum of finite arithmetic series; geometric sequences and series; sum of finite geometric series

2. Arithmetic Sequence

3. Arithmetic Sequences
An arithmetic sequence is a sequence in which each term differs from the pervious one by the same fixed number
Example
2,5,8,11,14
5-2=8-5=11-8=14-11 etc
31,27,23,19
27-31=23-27=19-23 etc

4. Algebraic Definition
{an} is arithmetic  an+1 – an= d for all positive integers n where d is a constant (the common difference)
 “If and only if”
{an} is arithmetic then an+1 – an is a constant and if an+1 – an is constant the {an} is arithmetic

5. The General Formula
a1 is the 1st term of an arithmetic sequence and the common difference is d
Then a2 = a1 + d therefore a3 = a1 + 2d therefore a4 = a1 + 3d etc.
Then an = a1 + (n-1)d
the coefficient of d is one less than the subscript

6. Find d for the following sequences
87, 83, 79, 75...
d=-4
8, 9.5, 11,
d=1.5
19, 25, 31, 37...
r=6
No common difference!
Arithmetic
Sequence
31, 37, 41, 47...

7. Example #1 Consider the sequence 2,9,16,23,30…
Show that the sequence is arithmetic
Find the formula for the general term Un
Find the 100th term of the sequence
Is 828, 2341 a member of the sequence?

8. Geometric Sequence

9. Have a common difference MULTIPLY to get next term
Arithmetic Sequences
Geometric Sequences
ADD To get next term
Have a common difference
MULTIPLY to get next term
Have a common ratio

10. Try to think of some geometric sequences on your own!
In a geometric sequence, the ratio of any term to the previous term is constant.
You keep multiplying by the SAME number
each time to get the sequence.
This same number is called the common ratio
and is denoted by r
What is the difference between an arithmetic sequence and a geometric sequence?
Try to think of some geometric sequences on your own!

11. Find r for the following sequences
4, 8, 16, 32...
r=2
r=3
8, 24, 72,
r=4
6, 24, 96,
No common ratio!
Geometric
Sequence
5, 10, 15, 20...

12. Writing a rule
To write a rule for the nth term of a geometric sequence, use the formula:

13. Writing a rule
Write a rule for the nth term of the sequence 6, 24, 96, 384,
Then find
This is the general rule. It’s a formula to use to find any term of this sequence.
To find , plug 7 in for n.

14. Writing a rule
Write a rule for the nth term of the sequence 1, 6, 36, 216, 1296,
Then find
This is the general rule. It’s a formula to use to find any term of this sequence.
To find , plug 8 in for n.

15. Writing a rule Write a rule for the nth term of the sequence
7, 14, 28, 56, 128,
Then find

16. (when you're not given the first term)
Writing a rule
(when you're not given the first term)
One term of a geometric sequence is The common ratio is r = Write a rule for the nth term.

17. (when you're given two non-consecutive terms)
Writing a rule
(when you're given two non-consecutive terms)
One term of a geometric sequence is and one term is
Step 1: Find r
-divide BIG
small
-find the distance between the two terms and take that root.
Step 2: Find Plug r, n, and into your equation. Then, solve for
Step 3: Write the equation using r and

18. (when you're given two non-consecutive terms)
Writing a rule
(when you're given two non-consecutive terms)
Write the rule when and

19. Series and Sequences Formulas

20. Graphing the sequence Let’s graph the sequence we just did.
Create a table of values. What kind of function is this?
What is a? What is b?
Why do we pick all positive whole numbers?
Domain, Input, X
Range, Output, Y

21. Compound Interest Formula
P dollars invested at an annual rate r, compounded n times per year, has a value of F dollars after t years.
Think of P as the present value, and F as the future value of the deposit.

22. Compound Interest
So if we invested $5000 that was compounded quarterly, at the end of a year looks like:
After 10 years, we have:

23. Series and Sequences Formulas