1. Warm Up
Evaluate each expression for x = 4, 5, 6.
1. 2x x + 1.5
3. x x + 3
9, 11, 13
3.5, 4, 4.5
19, 35, 67
15, 24, 35

2. Sequences & Series Day 1 Section 9.4 WHY?
Infinite sequences, especially those with finite limits are involved in some key concepts in calculus.

3. Notation & Vocabulary
A sequence is an ordered set of numbers. Each number in the sequence is a term of the sequence. A sequence may be an infinite sequence that continues without end, such as the natural numbers, or a finite sequence that has a limited number of terms, such as {1, 2, 3, 4}.

4. *** some books use n, some use k ***
Notation & Vocabulary
Instead of function notation, such as f(x) or a(n), sequence values are written by using subscripts. The first term is a1, the second term is a2, and the nth or kth term is an or ak. Because a sequence is a function, each number n or k has only one term
value associated with it, an or ak.
*** some books use n, some use k ***
an is read “a sub n.”
Reading Math

5. Defining a Sequence Explicitly
Example 1: Find the first 6 terms and the 100th term of the sequence {ak}, in which ak = k2 – 1 ak
Process: k2 – 1
Term a1 a2 a3 a4 a5 a6
a100 ak
Process: k2 – 1
Term a1
12 – 1
= 0 a2
22 – 1
= 3 a3
32 – 1
= 8 a4
42 – 1
= 15 a5
52 – 1
= 24 a6
62 – 1
= 35
a100
1002 – 1
= 9999
Go back to your warm up #3, does this match up for x = 4, 5, 6???

6. Example 1 Continued
You can use your calculator by using the function: y= x2 – 1
Go to table set and choose ASK for Independent
Then simply type the x values and press enter

7. Example 1 Continued
You can also use your calculator by typing the following commands:
0k
k+1k: k2 – 1
 is the STO> button
k is the ALPHA character for the ( button
: is the ALPHA character for the . Button
Every time you press enter it will show you the next term starting with a1

8. Defining a Sequence Recursively
Example 2: Find the first 6 terms and the 100th term of the sequence defined recursively by the conditions:
b1 = 3
bn = bn for all n > 1 bn
Process
Term b1
= 3 b2
b  b1 + 2  3+2
= 5 b3 b4 b5 b6
= 7
= 9
= 11
= 15

9. Defining a Sequence Recursively
Example 2: Find the first 6 terms and the 100th term of the sequence defined recursively by the conditions:
b1 = 3, bn = bn for all n > 1
Can you see that pattern or sequence?
{3, 5, 7, 9, 11, 13, …}
To get the 7th term, it is 6 terms beyond the 1st which means that we can quickly get there by adding 6 2’s to the 1st term, 3.
b7 = (2) = 15 Is this reasonable for the sequence?
Now find the 100th term:
201

10. Fibonacci Sequence
You have probably heard of the Fibonacci Sequence. One way to propagate the sequence is with the help of Pascal’s Triangle: 1
1 1
= 1
You can find each term of the sequence using addition, but the sequence is not arithmetic. The recursive formula for the Fibonacci sequence is:
Fn = fn-2 +Fn-1
F1 =1
F2 = 1
= 1
= 2
= 3
= 5
= 8

11. Arithmetic vs. Geometric
Graphs of Arithmetic Sequences relate to linear functions
(think adding for consecutive terms)
Graphs of Geometric Sequences relate to exponential functions
(think multiply for consecutive terms)

12. Arithmetic & Geometric Sequences
Recursive:
a1 = a given value
an = an-1 + d
(for all n > 2)
Explicit:
Recursive:
a1 = a given value
Explicit:
(for all n > 2)

13. Identify: Arithmetic or Geometric
Example 3: Identify the given sequences as arithmetic, geometric, neither or either?
Start by checking common differences and common ratios
-5, 10, -20, 40, …
2, 2, 2, 2, …
25, 50, 75, 100, …
1, 4, 9, 16, …
Geometric, the common ratio is -2
Either, the common difference is 0
And the common ratio is 1
Arithmetic, the common difference is 25
Neither, there are no common difference or common ratio

14. Use Means to find missing terms
The Arithmetic mean is simply adding 2 numbers and dividing by 2, it is what you think of when you think “average”

15. Use Means to find missing terms
Example 4: Find the missing term of the arithmetic sequence 84, ___, 110, …
Example 5: Find the missing term of the arithmetic sequence 24, ___, 57, …

16. Use Means to find missing terms
Example 6: Find the missing term of the geometric sequence 3, ___, 18.75, …
Example 7: Find the missing term of the geometric sequence 9180 , ___, 255, …

17. Writing Arithmetic Formulas
Example 8: Write a recursive and an explicit formula for the given arithmetic sequence:
-32, -20, -8, 4, 16
d, (common difference) = 12, a1, (1st term) = -32
Explicit:
an = a1 + (n-1)d
Recursive:
a1 = the first term
an = an-1 + d (for all n > 2)
an = (n-1)12
a1 = -32
an = an , (for all n > 2)

18. Writing Geometric Formulas
Example 9: Write a recursive and an explicit formula for the given geometric sequence:
3, 6, 12, 24, 48, …
r, (common ratio) = 2, a1, (1st term) = 3
Explicit:
an = a1 * r n-1
Recursive:
a1 = the first term
an = an-1 * r (for all n > 2)
an = (3) 2 n-1
a1 = 3
an = (2)an-1 (for all n > 2)

19. End Behavior—Limits of Sequences
Just as we are concerned with the end behavior of functions, we will also be concerned with the end behavior of sequences.
Converges
Diverges
If the degree of the numerator is the same as the degree of the denominator, the limit is the ratio of the leading coefficients
If the degree of the numerator is less than the degree of eh denominator, the limit is zero.
If the degree of the numerator is greater than the degree of the denominator, the limit is infinite.

20. Sequences and Series A sequence is a pattern
A series is the sum of the terms in the sequence
Finite Sequence
Finite Series
6, 9, 12, 15, 18
Infinite Sequence
Infinite Series
3, 7, 11, 15, … …

21. Arithmetic & Geometric Finite Series

22. Geometric Infinite Series
For an infinite Geometric series, use the following, where a1 is the 1st term and r is the common ratio.

23. End Behavior—Limits of Series
Just as we are concerned with the end behavior of functions, we will also be concerned with the end behavior of series.
Converges
Diverges, (no limit)
|r|<1
*converges to the sum S
|r|>1
*diverges, no limit

24. HW 9.4 p.739, 1-31 odd Project due 5/2 or 5/3
Quiz at the end of next class