Number Sequences and Series (Mathematic)

Number Sequences and Series

(Mathematic)

Wahyuni, S.Si., M.Sc.

Contents of This Class

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Arithmetic & Geometric progressions The Binomial Series

Trigonometry

Geometry Shapes

Arithmatic

In an Arithmetic Sequence the difference between one term and the next term is a constant.

Introduction

01

A sequence, terms., arithmetic sequence, and common difference.

A sequence is an ordered list of numbers usually separated by commas.

It contains elements or terms that follow a pattern or rule to determine the next term in the sequence.

The numbers in sequences are called terms.

An arithmetic sequence is an ordered list of terms in which the

difference between consecutive terms is a constant. The value added to each term to create the next term is the common difference.

2, 4, 6, 8, 10, 12  7, 3, -1, -5, -9 

common difference 14

-13

2 -4

The n value gives the relative position of each term.

1 2 3 4 5

3, 6, 9, 12, 15 ^{The tn} value gives the

actual terms of the sequence.

This is a finite arithmetic sequence

where

t

_n represents the

n

^th term of the sequence.

n

t

_n ^{n N}

The terms of a sequence are labelled according to their position in the sequence.

The first term of the sequence is t₁ or a.

The number of terms in the sequence can be represented by n.

The general term of the sequence (general rule) is t_n. This term is dependent on the value of n.

t

1

t

₂

t

₃

t

₄

t

₅

What assumptions are made?

What would change to write an infinite arithmetic sequence?

3, 6, 9, 12, 15,…

Given the sequence -5, -1, 3 …

b) Determine the value of the common difference.

d = t

₂

- t

₁

= ( -1) - ( -5) = 4

Note: the common difference may be found by subtracting any two consecutive terms.

c) What strategies could you use to determine thevalue of t

₁₀

?

Arithmetic Sequences

a) What is the value of t

₁

? -5 t

₃

? 3 t

₄

? 7

Deriving a Rule for the General Term of an Arithmetic Sequence

-5 -1 3 7

-5 -5 + (4) -5 + (4) + (4) -5 + (4) + (4) + (4)

t1 t₁  d t₁  d  d t₁  d  d  d

tn

-5 + (4) +… + (4)

1 ( 1)

t  n  d

An arithmetic sequence is a sequence that has a constant common difference, d, between successive terms.

t n = t 1 + ( n - 1) d.

General term or nth term

First term Position of

term in the sequence Common difference Terms

Sequence

Sequence Expressed using first term and common difference

General Sequence

t4

t3

t2

t1

The first term of an arithmetic sequence is (a)  We add, (d)  to get the next term. There is a pattern, therefore there is a formula we

can use to give use any term that we need without listing the whole sequence. This is called an explicit rule:

The n^th term of an arithmetic sequence is given by:

The value of the term

you are looking for First term The position the term is in (term

number)

The common difference

Determine the value of t_10.

t

_n

= t

₁

+ (n - 1) d

Write the expression for the general term.

t

_n

= t

₁

+ (n - 1) d

t₁ = -5 n = 10 d = 4 t₁₀ = ?

t₁ = -5 n = var d = 4

t

₁₀

= -5 + (10 - 1) 4 = -5 + (9) 4

t

₁₀

= 31

= -5 + (n - 1) 4 = -5 + 4n - 4 t

_n

= 4n - 9

parameters t₁ and d must be defined

t

₁₀

= 4(10) - 9 t

₁₀

= 40 - 9

t

₁₀

= 31

Explicit Definition

-5, -1, 3 …

Examples: Find the 14

^th

term of the arithmetic sequence 4, 7, 10, 13,……

(14 1)

 

4

4 (13)3

 

4 39

 

 43

a n = a 1 + (n – 1) d

a

₁₄

= 3

You are looking for

the term!

The 14

^th

term in this sequence is the number 43!

Examples:

Find the 14^th term of the arithmetic sequence with first term of 5 and the common difference is –6

.

(14 1)

 

a

_n

= a

₁

+ (n – 1) d

t

₁₄

=

You are looking for the term! List which variables

from the general term are provided!

The 14

^th

term in this sequence is the number -73!

a = 5 and d = -6

5 -6

= 5 + (13) (-6)

= 5 + -78

= -73

Examples: In the arithmetic sequence 4,7,10,13,…, which term has a value of 301?

301   4 ( n  1)3 301   4 3 n  3

301   1 3 n 300  3 n

100  n

You are

looking for n!

The 100

^th

term in this sequence is 301!

Arithmetic Sequence

Add some value each time on to infinity.

For example:

1, 4, 7, 10, 13, 16, 19, 22, 25, …

This sequence has a difference of 3 between each number.

It’s rule is

a

_n

= 3n – 2.

In general, we can write an arithmetic sequence like this:

a, a + d, a + 2d, a + 3d, …

a is the first term.

d is the difference between the terms (called the “common difference”)

The rule is:

Arithmetic Sequence

x

_n

= a + d(n-1)

(We use “n-1” because d is not used on the 1^st term.)

1.) 4,9,14,19, 24

First term (a): 4

Common difference (d):

^a₂ ^{ a}₁

= 9 – 4 = 5

2.) 34, 27, 20,13, 6, 1, 8,....  

First term (a): 34

Common difference (d): -7

Find the first term and the common difference of each:

Arithmetic Sequence

Arithmetic Sequence

For each sequence, if it is arithmetic, find the common difference.

-3, -6, -9, -12, …

1.1, 2.2, 3.3, 4.4, … 41, 32, 23, 14, 5, … 1, 2, 4, 8, 16, 32, …

1. d = -3 2. d = 1.1 3. d = -9

4. Not an arithmetic sequence.

Identify the first term and common difference, then

write a recursive rule for each of the sequences below:

Exercise

Find the 10^th and 25^th term given the following information.

Make sure to write the appropriate rule for the sequence to help you!!

a) 1, 7, 13, 19 ….

c) The second term is 8 and the common difference is 3 b) The first term is 3 and the common difference is -21

Example

Arithmetic Sequence

Write the explicit rule for the sequence

19, 13, 7, 1, -5, … Start with the formula:

x

_n

= a + d(n-1)

a is the first term = 19

d is the common difference = -6 The rule is:

x_n = 19 - 6(n-1)

Find the 12^th term of this sequence.

Substitute 12 in for “n”

x₁₂ = 19 - 6(12-1)

x₁₂ = 19 - 6(11)

x₁₂ = 19 – 66

x₁₂ = 19 - 6(12-1)

x₁₂ = -47

Write a recursive and an explicit rule for the sequences. Find the 30

^th

term

a) {16, 13, 10, 7, 4, …}

b) {-1.5, 1, 3.5, 6, 8.5, …}

Examples:

Are the following sequences arithmetic?

Explain.

1) 2, 5, 8 , 11,. . . .

2) 46, 47, 49, 51, . . . .

Examples:

Geometric

Sequences

Introduction

01

A sequence, terms., Geometric sequence, and common difference.

Geometric Sequence

In a Geometric Sequence each term is found by multiplying the pervious term by a constant.

For example:

2, 4, 8, 16, 32, 64, 128, …

The sequence has a factor of 2 between each number.

It’s rule is

x

_n

= 2

ⁿ

Geometric Sequence

In general we can write a geometric sequence like this:

a, ar, ar

²

, ar

³

, …

a is the first term

r is the factor between the terms (called the

“common ratio” ).

The rule is x

_n

= ar

^(n-1)

We use “n-1” because ar⁰ is the 1^st term.

𝑎

_𝑛

= Suku ke-n

𝑟 = 𝑎₂ 𝑎₁

Geometric Sequence

For each sequence, if it is geometric, find the common ratio.

2, 8, 32, 128, …

1, 10, 100, 1000, … 1, -1, 1, -1, …

20, 16, 12, 8, 4, …

1. r = 4 2. r = 10 3. r = -1

4. Not a geometric sequence.

Geometric Sequence

Write the explicit rule for the sequence 3, 6, 12, 24, 48, …

Start with the formula:

x

_n

= ar

^(n-1)

a is the first term: 3

r is the common ratio: 2 The rule is:

x_n = (3)(2)^(n-1)

Find the 12^th term of this

sequence. Substitute 12 in for “n.”

x₁₂ = (3)(2)^(12-1) x₁₂ = (3)(2)⁽¹¹⁾ x₁₂ = (3)(2048) x₁₂ = 6,144

(Order of operations states that we would take care of exponents before you multiply.)

The Binomial

Series

Pascal’s triangle

02

A binomial expression is one that contains two terms connected by a plus or minus sign. Thus (p+q),

(a+x)², (2x+y)³ are examples of binomial expression.

Expanding (a+x)ⁿ for integer values of n from 0 to 6 gives the results shown at below

Problem.

Determine, using Pascal’s triangle method, the expansion of

(2 p−3q)⁵

Comparing (2 p−3q)⁵ with (a+x)⁵ shows that a=2p and x=−3q Using Pascal’s triangle method:

(a + x)

5 = a⁵ +5a⁴x +10a³x² +10a²x³+…

Hence

(2p −3q)5 = (2p)⁵ +5(2p)⁴(−3q) +10(2p)³(−3q)²

+10(2p)²(−3q)3

+5(2p)(−3q)⁴ +(−3q)⁵ i.e. (2p−3q)⁵ = 32p⁵−240p4q+720p³q²

−1080p²q³+810pq⁴−243q⁵

The binomial series

The binomial series or binomial theorem

The binomial series or binomial theorem is a formula for raising a binomial expression to any power without lengthymultiplication.The general binomial expansion of (a+x)ⁿ is given by:

where, for example, 3! denote 3×2×1 and is termed ‘factorial 3’.

With the binomial theorem n may be a fraction, a decimal fraction or a positive or negative integer.

In the general expansion of (a+x)ⁿ it is noted that the 4th term is:

The number 3 is very evident in this expression.

If a=1 in the binomial expansion of (a+x)ⁿ then:

which is valid for −1< x < 1

When x is small compared with 1 then:

Problem.

Use the binomial series to determine the expansion of (2+x)⁷

The binomial expansion is given by:

When a=2 and n=7:

=

i.e.

(2+x)⁷ = 128+448x+672x²+560x³ +280x4+84x5+14x6+x⁷

Problem.

Evaluate (1.002)⁹ using the binomial theorem correct to (a) 3 decimal places and (b) 7 significant figures

Substituting x=0.002 and n=9 in the general expansion for (1+x)ⁿ gives:

Thank You

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Trigonometry

Trigonometry is the branch of mathematics that deals with the measurement of sides and angles of triangles, and their

relationship with each other. There are many applications in engineering where knowledge of trigonometry is needed.