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# 2.3 Sequences & Series to solve problems AS 91258

###### Due for update to realigned standard in 2013

 Sequences and Series involves number patterns: (1) Finding a particular number in the pattern (eg the 20th term) (2) Finding the sum of a section of the list of numbers (series) There are two general types of number patterns at level 2: (1) Arithmetic sequences (a common difference - adding or subtracting a constant) (2) Geometric sequences (a common multiplier - which can be a fraction) The sum of all numbers in a geometric sequence can be found if the common multiplier is between 0 and 1(sum to infinity)

#### 1

Overview

Overview of sequences and number patterns, Sequence notation using <> mapping notation and terms: t1, t2, t3, t4... continuing sequences using mapping notation.

Recursive sequences: continuing recursive sequences & forming a recursive formula.

Introduction to sequences

General sequences Ex25.01,
Recursive Sequences Ex25.02,

Class notes, Blank notes

Fibonacci Sequence at http://nlvm.usu.edu Explore the Fibonacci sequence and the golden ratio.

kovalevsy.pdf

Theta pg 117

#### 2

Sigma notation

Using sigma notation: Upper & lower limits, Terms, calculating a sum. Writing a sum in sigma notation.

Class notes, Blank notes

Sigma notation: Ex25.03

Theta pg 118

#### 3

Arithmetic Sequences

Continuing an arithmetic sequence with a first term 'a' and common DIFFERENCE 'd'.
Using the formula tn = a + (n - 1)d
Finding a specific term tn, finding a general term for an arithmetic sequence, calculating any one of 'a' 'd' 'n' and tn given three other values

Graphing arithmetic sequences.

Ex26.01

Theta pg 119

#### 4

Arithmetic Sequences (cont)

Calculating 'a' & 'd' using simultaneous equations given two terms

Ex26.02

Theta pg 120

#### 5

Arithmetic series

Finding the sum of an arithmetic sequence.
Using the formula Sn = n/2[2a + (n - 1)d]
Finding a specific sum Sn, calculating any one of 'a' 'd' 'n' and Sn given three other values

Ex26.03

Theta pg 120

#### 6

Arithmetic Applications

Using arithmetic sequences and series to solve application. Read the problem carefully, extract the important information: a=? d=? Sum or term? What is the question asking for?

Ex26.04

Theta pg 121 & 122

#### 7

Geometric Sequences

Continuing a geometric sequence with a first term 'a' and common RATIO 'r'.
Using the formula tn = a*r^(n - 1)
Finding a specific term tn, finding a general term for an arithmetic sequence, calculating any one of 'a' 'd' 'n' and tn given three other values

Graphing arithmetic sequences.

Ex27.01

Theta pg 123

#### 8

Geometric Sequences (cont)

Calculating 'a' & 'r' using simultaneous equations given two terms.

Using Logs to calculate 'n' given 'a' and 'r'

Ex27.02

Theta pg 124

#### 9

Geometric Series

Finding the sum of an geometric sequence.
Using the formula Sn = a(rn - 1)/(r-1)
Finding a specific sum Sn, calculating any one of 'a' 'r' 'n' and Sn given three other values

Ex27.03

Theta pg 125

#### 10

Geometric Applications

Using geometric sequences and series to solve application. Read the problem carefully, extrach the important information: a=? d=? Sum or term? What is the question asking for?

Ex27.04

Theta pg 126

#### 11

Sum to infinity

When a geometric series has a common ratio between -1 and 1 (exclusive) a sum to infinity is a finite value

Sinfinity = a / 1-r

Sum to infinity 2 (JW) and Mrs Adairs Sum to infinity

Sum to infinity powerpoint  (web version

Sum to infinity animation (maths on line)
Koch's snowflake - sum to infinity concept  | Image A | Image B | Image C

Class notes, Blank notes

Simple sum to infinity Ex27.05

Application problems Ex27.06

Theta pg 126

#### 12

Investment Applications

An investment of a Principle 'P' for 'n' years at '15%' increase interest rate = P*(1.15)n (or for inflation)

An investment of a Principle 'P' for 'n' years at '15%' decrease interest rate = P*(0.85)n (or for depreciation)

Ex28.01 Ex28.02 Ex28.03

Theta pg 127 & 128

#### 13

Revision

Maths tutorial websites Sequences, Series, Arithmetic, Geometric,

Revision test pg 411

Theta pg 129 & 130

TEST