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NZAMT NZQA NZ Grapher NZ Maths Census at School Study It

2.3 Sequences & Series to solve problems AS 91258

Sequences & Series HOME | Achievement Criteria | Revision

 

 

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)

Word doc 07/03 Theta Homework Book (David Barton) Theta Text Book (David Barton)
Web link Geogebra java animation Powerpoint Excel Spreadsheet (allow macros)

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.

Achievement Criteria

Introduction to sequences

Achievement criteria | I Can Do's

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

Class notes, Blank notes

Achievement criteria

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

Simplifying A

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.

Class notes, Blank notes

Ex26.01

Algebra Fractions

Theta pg 119

4



Arithmetic Sequences (cont)

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

Class notes, Blank notes

Ex26.02

Links to Guru maths website sequences, Arithmetic, Geometric, Convergence

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

Class notes, Blank notes

Ex26.03

Links to Maths tutorial websites Sequences, Series, Arithmetic, Geometric,

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?

Class notes, Blank notes

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.

Geometric sequences (featuring Bad Jelly)  (web version

Class notes, Blank notes

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'

Class notes, Blank notes

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

Geometric Series  (web version

Class notes, Blank notes

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?

Class notes, Blank notes

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)

Class notes, Blank notes

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

14



TEST

 

 

 

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