Warm-up: 1. For an arithmetic sequence, , find,

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

HW Solutions: WKS 1, 5, 9, 13, 17 ; Arithmetic ; d=4 2. d=-3 ; An=103-3n 3. 1,380 4. -578 5. 5201.5 6. 2,430

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

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

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

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.

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

Ex 1: Simplify

Ex 2: Simplify

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

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.

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.

Ex 4: Find the sum of the geometric series: 4 + 12 + ..... + 972. .

Ex 5: Determine the sum.

Ex. 6: Find the sum:

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

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?