Topic 1

ArithmeticSequences And

Series

Look at these number sequences carefully can you guess the next 2 numbers?

What about guess the rule?

30 40 50 60 70 80

17 20 292623 32

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48 41 34 27 20 13

+10

+3

-7

Can you work out the missing numbers in each of these sequences?

50

30

17515012510075

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50 70 90 110 130

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171176181186191196

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256266276286296306

+25

+20

-5

-10

Now try these sequences – think carefully and guess the last number!

1 2 164 7 11

3

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12 24 48 966

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0.5 2 3.5 5 6.5 8

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7 -5-214 -8

+1, +2, +3 …

double

+ 1.5

-3

This is a really famous number sequence which was discovered by an Italian mathematician a long time ago.

It is called the Fibonacci sequence and can be seen in many natural things like pine cones and sunflowers!!!0 1 1 2 3 5 8 13 21 etc…Can you see how it is made? What will the next number be?

34!

Guess my rule!

For these sequences I have done 2 math functions!

3 7 3115 63 127

2x - 1

2 3317953

2x + 1

What is a Number Sequence?

A list of numbers where there is a pattern is called a number

sequenceThe numbers in the sequence are

said to be its members or its terms.

SequencesSequences

To write the terms of a sequence given the nth term

Given the expression: 2n + 3, write the first 5 terms

In this expression the letter n represents the term number. So, if we substitute the term number for the letter n we will find value that particular term.

The first 5 terms of the sequence will be using values for n of: 1, 2, 3, 4 and 5

term 12 x 1 + 3

5

term 22 x 2 + 3

7

term 32 x 3 + 3

9

term 42 x 4 + 3

11

term 52 x 5 + 3

13

SequencesSequencesNow try these:

Write the first 3 terms of these sequences:

1) n + 2

2) 2n + 5

3) 3n - 2

4) 5n + 3

5) -4n + 10

6) n2 + 2

3, 4, 5

7, 9, 11

1, 4, 7

8, 13, 18

6, 2, - 2,

3, 6, 11,

6B - The General Term of A Number Sequence

Sequences may be defined in one of the following ways:

• listing the first few terms and assuming the pattern represented continues indefinitely

• giving a description in words

• using a formula which represents the general term or nth term.

The first row has three bricks, the second row has four bricks, and the third row has five bricks.

• If un represents the number of bricks in row n (from the top) then u1 = 3, u2 = 4, u3 = 5, u4 = 6, ....

This sequence can be describe in one of four ways:

• Listing the terms:

• u1 = 3, u2 = 4, u3 = 5, u4 = 6, ....

This sequence can be describe in one of four ways:

• Using Words: The first row has three bricks and each successive row under the row has one more brick...

This sequence can be describe in one of four ways:

• Using an explicit formula: un = n + 2u1 = 1 + 2 = 3u2 = 2 + 2 = 4u3 = 3 + 2 = 5u4 = 4 + 2 = 6, ....

This sequence can be describe in one of four ways:

• Using a graph

What you really need to know!

An arithmetic sequence is a sequence in which the difference between any two consecutive terms, called the common difference, is the same. In the sequence 2, 9, 16, 23, 30, . . . , the common difference is 7.

An An Arithmetic Arithmetic SequenceSequence is definedis defined as as

a sequence in which a sequence in which there is a there is a common common differencedifference between between consecutive terms.consecutive terms.

What you really need to know!

A geometric sequence is a sequence in which the quotient of any two consecutive terms, called the common ratio, is the same. In the sequence 1, 4, 16, 64, 256, . . , the common ratio is 4.

Example 1:

State whether the sequence -5, -1, 3, 7, 11, … is arithmetic. If it is, state the common difference and write the next three terms.

Example 2:

SubtractSubtract Common differenceCommon difference

11 – 7 11 – 7 447 – 3 7 – 3 443 – -1 3 – -1 44

-1 – -5 -1 – -5 44

-5, -1, 3, 7, 11,

Arithmetic! + 4

15, 19, 23

Example 2:

State whether the sequence 0, 2, 6, 12, 20, … is arithmetic. If it is, state the common difference and write the next three terms.

Example 2:

SubtractSubtract Common differenceCommon difference

20 – 12 20 – 12 8812 – 6 12 – 6 666 – 2 6 – 2 442 – 0 2 – 0 22

0, 2, 6, 12, 20 …

Not Arithmetic!

Example 3:

State whether the sequence 2, 4, 4, 8, 8, 16, 16 … is geometric. If it is, state the common ratio and write the next three terms.

Example 3:

DivideDivide Common ratioCommon ratio

16 16 ÷÷ 16 16 11

16 16 ÷÷ 8 8 22

8 8 ÷÷ 8 8 11

8 8 ÷÷ 4 4 22

4 4 ÷÷ 4 4 11

4 4 ÷÷ 2 2 22

2, 4, 4, 8, 8, 16, 16, …

Not Geometric!

Example 4:

State whether the sequence 27, -9, 3, -1, 1/3, … is geometric. If it is, state the common ratio and write the next three terms.

Example 4:

DivideDivide Common ratioCommon ratio

1/3 1/3 ÷÷ -1 -1 -1/3-1/3-1 -1 ÷÷ 3 3 -1/3-1/33 3 ÷÷ -9 -9 -1/3-1/3

-9 -9 ÷÷ 27 27 -1/3-1/3

27, -9, 3, -1, 1/3,

Geometric! • -1/3

-1/9, 1/27, -1/81

Which of the following sequences are arithmetic?

Identify the common difference.

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

15.5, 14, 12.5, 11, 9.5, 8, . . .

84, 80, 74, 66, 56, 44, . . .

8, 6, 4, 2, 0, . . .

50, 44, 38, 32, 26, . . .

YES 2d

YES

YES

NO

NO

1.5d

6d

The common

difference is

always the

difference between

any term and the

term that proceeds

that term.26, 21, 16, 11, 6, . . .

Common Difference = 5

The general form of an ARITHMETIC sequence.

1uFirst Term:

Second Term: 2 1 1u u d

Third Term:

Fourth Term:Fifth Term:

3 1 2u u d

4 1 3u u d

5 1 4u u d

nth Term: 1 1na a dn

Formula for the nth term of an ARITHMETIC sequence.

1 1nu u n d

nu th The n term

The term numbern

The common differenced

1u The 1st term

If we know any

If we know any threethree of these of these we ought to be

we ought to be able to find the

able to find the fourth.fourth.

Given: 79, 75, 71, 67, 63, . . .Find: 32u

1 79

4

32

u

d

n

IDENTIFY SOLVE

1 ( 1)nu u n d

32 79 (32 1)( 4)u

32 45u

Given: 79, 75, 71, 67, 63, . . .

Find: What term number is (-169)?

1 79

4

169n

u

d

u

IDENTIFY SOLVE

1 ( 1)nu u n d

)4)(1(79169 n

63nIf it’s not an integer, it’s not a term in the

sequence

Homework

Page 156 2 - 11( Any 8 Problems)

Take Home Test Due Tuesday.