THE BEST CLASS EVER…ERRR….PRE-CALCULUS

Chapter 13 Final Exam Review

Chapter 13 Previous Lesson Materials

What was Chapter 13 about?

Sequences Arithmetic Geometric Recursive

Series Finite Arithmetic Finite Geometric Infinite Geometric Sigma Notation

Sequences - Arithmetic

Sequence A list of numbers that follow a specific

pattern and the input values are from the set of positive integers

Arithmetic Sequence

= first term in the sequence n = “nth” term of the sequence d = common difference

Arithmetic Sequence- Example

Write a rule for the nth term of the arithmetic sequence: a5 = 18, a17 = 66

an = 4n – 2

Sequences - Geometric

Geometric Sequence

= first term of the sequence n = “nth” term of the sequence r = common ratio

Geometric Sequence- Example

Write a rule for the nth term of the geometric sequence: a2 = 6, a5 = 162

an =

Sequences - Recursive

Recursive Sequence Each following term of the sequence is dependent upon

knowing the value of the previous term2 Key Elements of a Recursive Sequence

An initial condition that tells where the sequence begins A recursive equation that describes how the previous

term is related to the next termExample:

23, 20, 17, 14, …

Recursive Sequence- Example

Write a recursive definition for the following sequences:

a) 9, 13, 17, 21, … b) 1, 3, 7, 13, 21, 31, …

a) tn = tn-1 + 4 b) tn = tn-1 + 2(n-1)

Series – Finite Arithmetic

Series: an indicated sum of the terms of a sequence Example of Sequence: 2, 4, 6, 8, … Example of Series: 2 + 4 + 6 + 8 + …

Finite Arithmetic Series

= sum of the first “n” terms n = number of terms = first term of the arithmetic SEQUENCE = “nth” term of the arithmetic SEQUENCE (reference

the formula for arithmetic sequence)

Series – Finite Geometric

Finite Geometric Series

= sum of the first “n” terms n = number of terms = first term of the geometric SEQUENCE = “nth” term of the geometric SEQUENCE (reference

the formula for geometric sequence)

Finite Series- Example #1

Find the sum of the first 15 terms of the geometric series: 5 – 15 + 45 – 135 + …

S15 = 179, 936, 135

Finite Series- Example #2

Find n if Sn = -10,602 for the arithmetic series 12 + 6 + 0 + …

n = 62

Series – Infinite Geometric

Infinite Geometric Series If |r|, then the geometric series converges to a number

that can be found by the following formula: S: the sum of the infinite geometric series = the first term of the geometric sequence r = the common difference

If |r| , then the geometric series diverges towards some type of infinity

Infinite Series- Example

Determine if the following infinite series diverges or converges. If it converges, state the sum.

a) Diverges b) S = 2

∑𝑛=1

∞

(−1)( 32)𝑛−1a) b)

∑𝑛=1

∞

( 12)( 34)𝑛−1

Series – Sigma Notation

Sigma Notation Represents a series in abbreviated form

The picture below describes the pieces of sigma notation.

Key points to recognize “4n” is the sequence that will develop the terms Don’t forget to include “+” in between the terms If the last value of n = , this signifies some type of infinite series

Sigma Notation- Example

What is the summation notation for the series 2 + 4 + 8 + 16?

∑𝑘=1

4

2𝑘