1. Warm-up: 1. For an arithmetic sequence, , find,
the recursive definition, and
the explicit definition

2. HW Solutions: WKS 1, 5, 9, 13, 17 ; Arithmetic ; d=4
2. d=-3 ; An=103-3n
3. 1,380
6. 2,430

3. Factorial Notation & Geometric Series
Unit 1 Chapter 11
Factorial Notation & Geometric Series

4. Objectives & HW: The students will be able to:
Evaluate expressions involving factorials
Find the partial sum of a geometric series
HW: p. 788: 22, 26, 30
p. 791: 2, 4, 5, 6

5. For any positive integer n,
n! = n(n – 1)(n-2) (3)(2)(1)
And 0! = 1
Why is 0! = 1?

7. A factorial can be defined recursively as:
Can factorials also be computed for non-integer numbers? Yes, there is a famous function, the gamma function G(z), which extends factorials to real and even complex numbers. This is a topic for more advanced mathematics courses.

8. Factorial
Evaluate:
1) 7! 2) 2!3!

9. Ex 1: Simplify

10. Ex 2: Simplify

11. Ex 3: Write the first four terms of the following sequence:

12. Deriving the formula for the sum of first n terms of a geometric sequence:
Write the sum.
Re-express each term of the sum. (equation 1)
Multiply both sides of equation 1 by r. (equation 2)
Subtract equation 2 from equation 1.
Factor.
Solve for Sn.

13. Explicit Formula for the partial sum of a geometric series:
where n is the number of terms,
a1 is the first term, and
r is the common ratio.

14. Ex 4: Find the sum of the geometric series: .

15. Ex 5: Determine the sum.

16. Ex. 6: Find the sum:

17. Ex. 6: Find the sum:
-410/729

18. Ex. 7: A ball is dropped from a height of 10 feet
Ex. 7: A ball is dropped from a height of 10 feet. It hits the floor and bounces to a height of 7.5 feet. It continues to bounce up and down. On each bounce it rises to ¾ of the height of the precious bounce. How far has it traveled (both up and down) when it hits the floor for the ninth time?