Arithmetic and geometric sequences SAMPLE

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CHAPTER

9

9.1

geoAmriethtMrmiOcDUesLtEei1cquaenndcesE What is an arithmetic sequence? L What is a geometric sequence?

How do we find the nth term of an arithmetic or geometric sequence? How do we find the sum of the first n terms of an arithmetic or geometric sequence? How do we find the sum to infinity of a geometric sequence? How can we use arithmetic and geometric sequences to model real-world

P situations?

How do we distinguish graphically between an arithmetic and a geometric sequence?

Sequences M We call a list of numbers written down in succession a sequence; for example, the numbers

drawn in a lottery: 12, 22, 5, 6, 16, 43, . . .

For this sequence, there is no clear rule that will enable you to predict with certainty the next

Anumber in the sequence. Such sequences are called random sequences. Random sequences

arise in situations where chance plays a role in determining outcomes, for example the number of people in successive cars arriving at a parking lot, or the number generated in a Tattslotto draw.

SIf, however, we list house numbers on the left-hand side of a suburban street:

1, 3, 5, 7, 9, 11, . . .

we see they also form a sequence. This sequence differs from the random sequence above in that there is a clear pattern or rule that enables us to determine how the sequence will continue; each term in the sequence is two more than its predecessor. Such a sequence is called a rule-based sequence. In this module, we will concentrate on rule-based sequences.

Cambridge TI-Nspire &

University Press Casio ClassPad

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?

Jones,

Evans,

Lipson

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Exercise 9A

1 Label each of the sequences as either rule based or probably random. If rule based, write

down the next value in the sequence.

a 1, 2, 3, 4, . . .

b 100, 99, 98, 97, . . .

c 23, 9, 98, 1, . . .

d 1, 1, 1, 0, 1, 0, 0, 0, . . . e 1, 1, 1, 1, . . .

g 2, 4, 6, 8, . . .

h 2, 4, 8, 16, . . .

f 1, 0, 1, 0, 1, 0, . . . i 10, 20, 40, 80, . . .

2 For each of the following rule-based sequences, find the missing term:

a 1, 10, 100, , . . .

d

1 7

,

2 7

,

3 7

,

,...

g

1 2

,

1 4

,

1 16

,

,...

b 10, , 6, 4, 2, . . . e 51, 52, , 58, . . . h 10, -10, , -10, . . .

c 3, 9, , 81, . . . f -2, -6, , -14, . . . i 2, -2, , -10, . . .

Arithmetic sequences E 9.2 L A sequence in which each successive term can be found by adding the same number is called

an arithmetic sequence. For example, the sequence 2, 7, 12, 17, 22, . . . is arithmetic because each successive term

can be found by adding 5.

term 1 term 2 term 3 term 4 term 5

P 2

7

12

17

22

+5

+5

+5

+5

+5

The sequence 19, 17, 15, 13, 11, . . . is also arithmetic because each successive term can be found by adding -2, or equivalently, subtracting 2.

M term 1 term 2 term 3 term 4 term 5

19

17

15

13

11

?2

?2

?2

?2

?2

AThe common difference

Because of the way in which an arithmetic sequence is formed, the difference between successive terms is constant. In the language of arithmetic sequences, we call it the common

Sdifference. In the sequence

2, 7, 12, 17, 22, . . .

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the common difference is +5, while in the sequence 19, 17, 15, 13, 11, . . .

the common difference is -2. Once you know the first term in an arithmetic sequence and its common difference, the rest

of the terms in the sequence can be readily generated. If you want to generate a large number of terms, your graphics calculator will do this with little effort.

How to generate the terms of an arithmetic sequence using the TI-Nspire CAS

Generate the first five terms of the arithmetic sequence: 2, 7, 12, 17, 22, . . .

E Steps

1 Start a new document by pressing / + N.

2 Select 1:Add Calculator.

L Enter the value of the first term, 2. Press

. enter 3 The common difference for the

sequence is 5. So, type +5. Press enter . The second term in the sequence, 7, is

P generated.

4 Pressing enter again generates the next term, 12.

5 Keep pressing enter until the desired

SAM number of terms is generated.

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How to generate the terms of an arithmetic sequence in the main application using the ClassPad

Generate the first five terms of the arithmetic sequence: 2, 7, 12, 17, 22, . . .

Steps 1 From the application menu

screen, locate the built-in Main application. Tap to open, giving the screen shown opposite. Note: Tapping from the icon panel (just below the

E touch screen) will display

the application menu, if it is not already visible. 2 a Starting with a clean

L screen, enter the value of

the first term, 2. b The common difference

for this sequence is 5. So, type +5. Then press E.

P The second term in the

sequence (7) is displayed. 3 Pressing E again

generates the next term, 12. Keep pressing E until the

M required number of terms is SAgenerated.

Being able to recognise an arithmetic sequence is another skill that you need to develop. The key idea here is that the successive terms in an arithmetic sequence differ by a constant amount (the common difference).

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Example 1

Testing for an arithmetic sequence

a Is the sequence 20, 17, 14, 11, 8, . . . arithmetic?

Solution

Strategy: Subtract successive terms in the sequence to see whether they differ by a constant

amount. If they do, the sequence is arithmetic.

1 Write down the terms of the sequence.

20, 17, 14, 11, 8, . . .

2 Subtract successive terms.

17 - 20 = -3

14 - 17 = -3

3 Write down your conclusion

11 - 14 = -3 and so on Sequence is arithmetic as terms differ by a

E constant amount.

b Is the sequence 4, 8, 16, 32, . . . arithmetic?

Solution

L 1 Write down the terms of the sequence.

2 Subtract successive terms.

P 3 Write down your conclusion

4, 8, 16, 32, . . . 8-4=4 16 - 8 = 8 32 - 16 = 16 and so on

Sequence is not arithmetic as terms differ by different amounts.

Exercise 9B 1 Which of the following sequences are arithmetic?

M a 1,2,3,4,...

d 1, 1, 0, 1, . . . g 6, 24, 36, 48, . . . j -10, -6, -2, 2, . . . m -100, -200, -400, . . .

b 100, 99, 98, 97, . . . e 3, 9, 27, . . . h 2, 4, 8, 16, . . . k 100, 50, 0, -50, . . . n 1, -3, -7, -11, . . .

c 23, 25, 29, 31, . . . f 56, 60, 64, . . . i 10, 20, 40, 80, . . . l -20, -15, -10, -5, . . . o 2, 4, 16, 256, . . .

A2 Which of the following sequences are arithmetic?

a 1.1, 2.1, 3.1, 4.1, . . . d 1.1, 1.1, 0.1, 1.1, . . .

Sg 6000, 7450, 8990,...

b 9.8, 8.8, 7.8, 6.8, . . . e 36.3, 37.0, 37.7, . . . h 2000, 4000, 8000, . . .

c 1.01, 1.02, 2.02, . . . f 0.01, 0.02, 0.04, . . . i 10.9, 20.8, 30.7, . . .

j -100, -60, -20, 20, . . . k -10.2, -5.1, 0, 5.1, . . . l -67.8, -154.1, -205.7, . . .

3 Consider the sequence 20, 24, 28, 32, 36, . . .

a Why is the sequence arithmetic? b What is the common difference? c What is the next term in the sequence?

Cambridge University Press ? Uncorrected Sample pages ? 978-0-521-61328-6 ? 2008 ? Jones, Evans, Lipson TI-Nspire & Casio ClassPad material in collaboration with Brown and McMenamin

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