SEQUENCES & SERIES

Sequences, Series, Sigma Notation, Limits, FTC

Introduction

DIFFERENCE BETWEEN SEQUENCE AND SERIES

2 2,1 8,1 4,1 0,6,2

This is a sequence: This is a series:

2 21 81 41 062 A sequence is just a list of numbers A series is a list of numbers being

added together

They can be as long or short as you want, some can even go on forever

Each number listed is called a TERM

The first term, the second term etc.We say nth term when referring to terms in general

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Sequences, Series, Sigma Notation, Limits, FTC

Introduction

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 …

The first term is 1

:nt

11 tThe second term is 1 12 tThe third term is 2 23 tThe seventh term is 1 37 t13

Here is a sequence:

NOTATION

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13

2

1

1

4

3

2

1

a

a

a

a

Some books use “t” for term, others use “a”.

Sequences, Series, Sigma Notation, Limits, FTC

Introduction

A Sequence is just a list of numbers in a particular order

A typical sequence question looks like this:

Find the next 2 numbers in these sequences

_ __ _ ,,2 6,2 0,1 4,8 _ __ _ ,,1 3 5,4 5,1 5,5

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Sequences, Series, Sigma Notation, Limits, FTC

Introduction

While finding the next number in a pattern is easy for many sequences, some are not so obvious:

How to identify the pattern and find the next number in a sequence:

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__,182,102,50,20,6,2

What do you add to 2 to get to 6?

4 14 30 52 80

Sometimes this makes it easier to see a pattern, if not, do it again!

10 16 22 286 6 6

Then work backwards to get the next number

6

34114

296

Sequences, Series, Sigma Notation, Limits, FTC

Introduction

Most of these problems are easier:

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___,162,54,18,6,23

4863 3 3 3

___,51,40,29,18,711

6211 11 11 11

All the problems we do will be… + a number (arithmetic)ORx a number (geometric)

Sequences, Series, Sigma Notation, Limits, FTC

Introduction

Can you figure this out without dividing?

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___,2

1,1,2,4,8,16,32,64

2

1

4

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

• You should now be able to do PAGE #2 part 1 in the yellow booklet:

Sequences, Series, Sigma Notation, Limits, FTC

basics

Every sequence can be described with an equation:

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32 na n53121 a

73222 a

93323 a

. . .,1 7,1 5,1 3,1 1,9,7,5

Plug numbers in for “n” to find the terms of the sequence.

To find the first 5 terms…plug in 1 – 5

If you wanted to know the 27th term, plug in 27.

Sequences, Series, Sigma Notation, Limits, FTC

basics

Find the first 5 terms of this sequence:

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4 1na n

1 4 1 1a 32 4 2 1a 7

3 4 3 1a 11

4 4 4 1a 15

3, 7 , 11, 15 , 19

Sequences, Series, Sigma Notation, Limits, FTC

basics

Find the first 4 terms of this sequence:

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2nt n n

21 1 1t 0

22 2 2t 2

23 3 3t 6

24 4 4t 12

0 , 2 , 6 , 12

(You can cheat by using the table feature in your calculator)

Sequences, Series, Sigma Notation, Limits, FTC

basics

Finding the pattern for basic sequences:

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__,625,256,81,16,1,0.5

__,57,34,17,6,1.4

__,14,10,7,5,4.3

__,40,20,10,5.2

__,27,20,13,6.1

Find the next term in each of the following sequences:

77734

2 01 05 22280

19

86

12 9 6

6 7 ( 1)nt n

( 1)5 2 nnt

20 .5 0 .5 4na x x

23 4 2nt x x

4( 1)na n

You should now be able to do PAGE#2, part 2 in the yellow booklet.

Sequences, Series, Sigma Notation, Limits, FTC

Sequences Menu

SEQUENCES

A Sequence is a list of numbers in a particular order.

We are going to look at 2 types of sequences: Arithmetic and Geometric

2 2,1 8,1 4,1 0,6,2

A sequence is ARITHMETIC if:

Each term goes up/down by the same amount44444

1 8,2 1,2 4,2 7,3 03333

The number it goes up/down byIs called the common difference

Sequences, Series, Sigma Notation, Limits, FTC

Sequences Menu

,1 6 2,5 4,1 8,6,2

A sequence is GEOMETRIC if:

Each term is multiplied/dividedby the same amount3333

4,8,1 6,3 2,6 41 1 1 12 2 2 2

The number it is multipliedby is called the common ratio

Sequences, Series, Sigma Notation, Limits, FTC

Sequences Menu

You should now be able to do PAGE#3 in the yellow booklet.

FORMULAS FOR THE nth TERM

,1 4,1 1,8,5,2

ARITHMETIC: 1st term

)1(1 ndtt n

nth term

The common difference)1(32 nt n

Put 4 in:

Out comes 11, the 4th term!

Sequences, Series, Sigma Notation, Limits, FTC

Sequences Menu

FORMULAS FOR THE nth TERM

,3 3,2 6,1 9,1 2,5

ARITHMETIC:

)1(75 nt n

Put 4 in:

Out comes 26, the 4th term!

Sequences, Series, Sigma Notation, Limits, FTC

Sequences Menu

1st term

)1(1 ndtt n

nth term

The common difference

FORMULAS FOR THE nth TERM

,1 8,2 1,2 4,2 7,3 0

ARITHMETIC:

)1(33 0 nt n)1(33 0 nt n

Put 4 in:

Out comes 21, the 4th term!

Sequences, Series, Sigma Notation, Limits, FTC

Sequences Menu

1st term

)1(1 ndtt n

nth term

The common difference

FORMULAS FOR THE nth TERM

,1 6 2,5 4,1 8,6,2

GEOMETRIC: 1st term

)1(1

nn rtt

nth term

The common ratio

)1(32 nnt

Put 4 in:

Out comes 54, the 4th term!

Sequences, Series, Sigma Notation, Limits, FTC

Sequences Menu

FORMULAS FOR THE nth TERM

,8 0,4 0,2 0,1 0,5

GEOMETRIC:

)1(25 nnt

Put 4 in:

Out comes 40, the 4th term!

Sequences, Series, Sigma Notation, Limits, FTC

Sequences Menu

1st term

)1(1

nn rtt

nth term

The common ratio

FORMULAS FOR THE nth TERM

21,1,2,4,8

GEOMETRIC:

)1(28 nnt

Put 4 in:

Out comes 1, the 4th term!

)1(

2

18

n

nt

Sequences, Series, Sigma Notation, Limits, FTC

Sequences Menu

1st term

)1(1

nn rtt

nth term

The common ratio

FORMULAS FOR THE nth TERM

21,1,2,4,8

)1(28 nnt

)1(

2

18

n

nt

Sequences, Series, Sigma Notation, Limits, FTC

Sequences Menu

These are called EXPLICIT definitions.

Next, we will talk about other ways to describe a sequence. (RECURSIVE definitions)

You should now be able to do PAGE#4 in the yellow booklet.