PATTERNS, SEQUENCES & SERIES (LIVE) 07 APRIL 2015 Section ...

PATTERNS, SEQUENCES & SERIES (LIVE) Section A: Summary Notes and Examples

07 APRIL 2015

Grade 11 Revision

Before you begin working with grade 12 patterns, sequences and series, it is important to revise what you learnt in grade 11 about quadratic sequences. A quadratic sequence is a sequence in which the second difference is constant. The general term of this sequence is = 2 + + =

=

=

Example

Consider the pattern: 5; -2; -7; -10; ...

1.

Write down the next two terms

2.

Determine an expression for the nth terms

3.

Show that the sequence will never have a term with a value less than -11

Solutions

1.

-11; -10

2.

Begin by identifying the sequence. Since the sequence doesn't have a common first

difference or a constant ratio, we check to see if the sequence is quadratic.

= 2 = 1

To find and substitute = 1 into = 2 + +

Equation 1 1 = 1 1 2 + + 5 - 1 = + 4 = +

Now substitute = 2

Equation 2 2 = 1 2 2 + (2) + -2 = 1 2 2 + 2 + -6 = 2 +

Now solve equation 1 and 2 simultaneously

Equation 2 minus equation 1 -10 = 4 = -10 + 14 = = 2 - 10 + 14

Page 1

3.

2 - 10 + 14 < -11

2 - 10 + 25 < 0

( - 5)2 < 0

This is not true for any values of thus the sequence will not have a term less than -11

Arithmetic Sequences and Series

An arithmetic sequence or series is a linear number pattern in which the first difference is constant.

The general term formula allows you to determine any specific term of an arithmetic sequence. And the sum of formula determines the sum of a specific number of terms of an arithmetic series.

The formulae are as follows:

= + - 1

=

2

[2

+

- 1

]

=

2

[

+

]

where = and = where = and = where is the last term

Note:

= 2 - 1 1 = 2 = +

3 = + 2 .

Example 1

The 19th term of an arithmetic sequence is 11, while the 31st term is 5.

(a) Determine the first three terms of the sequence. 19 = + 18 = 11 31 = + 30 = 12 = -6 = - 1

2

+ 18 -1 = 11

2

= 20 20; 19 1 ; 19 ...

2

(b) Which term of the sequence is equal to -29? = -29 = + ( - 1) 20 + - 1 - 1 = -29

2

- 1 - 1 = -49

2

- 1 = 98 = 99 99 = -29

Page 2

Example 2

Given: 1 + 2 + 3 + 4 + ... ... ... ... + 180

181 181 181 181

181

(a) Calculate the sum of the given series.

1 + 2 + 3 + 4 + ... ... ... ... + 180

181 181 181 181

181

= 1 = 1 = 180

181

181

181

180

=

180 2

2

1 181

+

179 1

181

= 90 1

= 90

(b) Hence calculate the sum of the following series:

1 + 1 + 2 + 1 + 2 + 3 + ... ... . . + 1 + 2 + ... ... . + 180

2

33

444

181 181

181

1 + 1 + 2 + 1 + 2 + 3 + ... ... . . + 1 + 2 + ... ... . + 180

2

33

444

181 181

181

= 1 + 1 + 1 1 + 2 ... ... ... . +90 [ = 1

2

2

2

= 1

2

= 90]

1 + - 1 1 = 90

2

2

1 + - 1 = 180

= 180

180

=

180 2

1 + 90

2

= 90

90 1

2

= 8145

Geometric Sequences and Series

A geometric sequence or series is an exponential number pattern in which the ratio is constant.

The general term formula allows you to determine any specific term of a geometric sequence. You have also learnt formulae to determine the sum of a specific number of terms of a geometric series.

The formulae are as follows:

= -1

( - 1) = - 1 where 1

=

1 2

1 = 2 = 3 = 2 .

Page 3

Example 1

In a geometric sequence in which all terms are positive, the sixth term is 3 and the eighth term is 27. Determine the first term and constant ratio.

6 = 3 and 8 = 27 5 = 3 7 = 27

7 27 5 = 3 2 = 27

3 2 = 9 2 = 3 2 = 3 (terms are positive) ( 3)5 = 3

3 =

( 3)5

1 =

( 3)4

=

1 (3 1)4

2

1 = 9

Convergent Geometric Series

Consider the following geometric series:

111 1 2 + 4 + 8 + 16 + ... ....

We can work out the sum of progressive terms as follows:

1

=

1 2

=

0,5

2

=

1 2

+

1 4

=

3 4

=

0,75

3

=

1 2

+

1 4

+

1 8

=

7 8

=

0,875

4

=

1 2

+

1 4

+

1 8

+

1 16

=

15 16

=

0,9375

(Start by adding in the first term) (Then add the first two terms) (Then add the first three terms) (Then add the first four terms)

If we continue adding progressive terms, it is clear that the decimal obtained is getting closer and closer to 1. The series is said to converge to 1. The number to which the series converges is called the sum to infinity of the series.

There is a useful formula to help us calculate the sum to infinity of a convergent geometric series.

The formula is

=

1-

If we consider the previous series 1 + 1 + 1 + 1 + ... ....

2 4 8 16

Page 4

It is clear that = 1 and = 1

2

2

= 1 -

1

=

1

2

-

1

=

1

2

A geometric series will converge only if the constant ratio is a number between negative one and positive one.

In other words, the sum to infinity for a given geometric series will exist only if -1 < < 1

If the constant ratio lies outside this interval, then the series will not converge.

For example, the geometric series 1 + 2 + 4 + 8 + 16 + ... ... ... ... will not converge since the sum of the progressive terms of the series diverges because = 2 which lies outside the interval -1 < < 1

Example 1

Given the geometric series: 82 + 43 + 24 + ...

(a)

Determine the nth term of the series.

= -1

= (82)

1 2

-1

(b) For what value(s) of will the series converge? -1 < < 1

2

= -2 < < 2

(c)

Calculate the sum of the series to infinity if = 3

2

=

1-

=

8 2 1-2

=

8(32)2 1-12(32)

= 72

Sigma Notation

Sigma means sum of, for example

6 =2

+

1

means

the

sum

of

the

five

terms

in

the

sequence

n+1.

We determine the number of terms in this sequence by subtracting the number at the bottom, 2, from

the number at the top, 6, and as seen below. There are 5 terms in the sequence.

6

+ 1 = 2 + 1 + 3 + 1 + 4 + 1 + 5 + 1 + [6 + 1]

=2 6

+ 1 = 3 + 4 + 5 + 6 + 7 = 25

=2

Page 5

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download