Sequences Series

Sequence & Series - Arithmetic

Sequences & Series

Arithmetic

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Sequences and Series: Arithmetic Sequences and series of numbers occur frequently in real life and mathematics. This booklet introduces the basic concepts focusing on arithmetic sequences and series. Several applications will also be discussed Answer these questions, before working through the chapter.

I used to think: What is the difference between a sequence and a series?

What is an arithmetic sequence?

What is the formula for the general term of an arithmetic sequences?

Suggest one practical application for the sum of an arithmetic series.

Answer these questions, after working through the chapter.

But now I think: What is the difference between a sequence and a series?

What is an arithmetic sequence?

What is the formula for the general term of an arithmetic sequences?

Suggest one practical application for the sum of an arithmetic series.

What do I know now that I didn’t know before?

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Sequences and Series: Arithmetic

Basics

Sequences and Terms A sequence is a list of numbers in a specific order like this This sequence has 4 numbers. The number of terms of the sequence is 4.

3, 5, 8, 9 Each number in a sequence is called a term of the sequence.

Above, the 1st term is 3 and the 2nd term is 5. We write this as T1 = 3 , T2 = 5 and so on. Here are two examples. Look at these sequences. a

b

4, 7, 34, 0, 9, 7

-10, 14, 20, -22, 26

T1 = 4

first term

T1 =-10 first term

T2 = 7

second term

T2 = 14

second term

T3 = 34 third term

T3 = 20

third term

T4 = 0

T4 =-22 fourth term

fourth term

The number of terms of the sequence is 6.

The number of terms of the sequence is 5.

General Term of a Sequence T1 is the 1st term of the sequence, T2 is the 2nd term of the sequence. Tn is the n th term of the sequence, also called the General term of the sequence. Here are two examples. Write down the first 5 terms of these sequence with the formulas given. a

The n th term of a sequence is given by the formula Tn = 4n + 5 . Use this with n = 1, 2, 3, 4, 5 to write down the first 5 terms of this sequence.

The n th term of a sequence is given by the formula Tn = 9 - 5n . Use this with n = 1, 2, 3, 4, 5 to write down the first 5 terms of this sequence.

T1 = 4^1 h + 5 = 9

T1 = 9 - 5 ^1 h = 4

T3 = 4^3 h + 5 = 17

T3 = 9 - 5 ^3 h =-6

T2 = 4^2h + 5 = 13

T2 = 9 - 5 ^2h =-1

T4 = 4^4h + 5 = 21

T4 = 9 - 5 ^4h =-11

T5 = 4^5 h + 5 = 25

T5 = 9 - 5 ^5 h =-16

` The first 5 terms are 9, 13, 17, 21, 25.

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` The first 5 terms are 4, -1, -6, -11, -16.

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Basics

Here are two examples where you need to find the formula for the general term of a sequence. Find the formula for the general term Tn for the sequences. a

5, 12, 19, 26, 33, 40, ...

b

The terms are increasing by 7. T1 = 5

125, 115, 105, 95, 85, ... The terms are decreasing by 10. T1 = 125

= 5+7#0

= 125 - 10 # 0

= 5+7#1

` T2 = 115 = 125 - 10 = 125 - 10 # 1

T3 = 19 = 5 + 14 = 5 + 7 # 2

T3 = 105 = 125 - 20 = 125 - 10 # 2

T4 = 26 = 5 + 21 = 5 + 7 # 3 ...

T4 = 95 = 125 - 30 = 125 - 10 # 3 ...

Tn = 5 + 7 # ^n - 1h

Tn = 125 - 10 # ^n - 1h

Tn = 5 + 7n - 7

Tn = 125 - 10n + 10

` T2 = 12 = 5 + 7

` Tn = 135 - 10n

` Tn = 7n - 2

Here is an example where a term is given and the n value found. The n th term of a sequence is given by Tn = 9n + 8 . For which value of n does the n th term equal 161? Substituting into the formula gives 161 = 9n + 8 ` 9n = 161 - 8 9n = 153 n = 153 9 n = 17

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Sequences and Series: Arithmetic

Questions

Basics

1. For the sequences shown, fill in the values indicated a

b

3, 6, 1, -7, 0, 45

9, 6, 2, -3, -10, 20, 4

The number of terms in the sequence =

The number of terms in the sequence =

T1 =

T1 =

T2 =

T2 =

T3 =

T3 =

T4 =

T4 =

2. Write down the first 5 terms of the sequence whose general term is given: a

Tn = 7n

b

Tn = 6n + 7

c

Tn = 1 - 3n

d

Tn =-19n

3. What is the 5 th term of the sequence with n th term:

4.

a

Tn = 100 + 8n

b

Tn =-25 - 6n

a

The n th term of a sequence is given by Tn = 8n - 5 . For which value of n does the n th term equal 163?

b

The n th term of a sequence is given by Tn = 3 - 12n . For which value of n does the n th term equal -177?

b

4, 7, 10, 13, ...

5. Try and find the general term, Tn , of these sequence a

4

2, 4, 6, 8, ...

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Sequences and Series: Arithmetic

Basics

Arithmetic Sequences An Arithmetic Sequence is a specific type of sequence where the difference (d) between consecutive terms is constant (i.e. the same) ` T1 , T2 , T3 , T4 , ... d

d

d

d = T2 - T1 = T3 - T2 = ... = Tn - Tn - 1 = ...

Always add or subtract the same number.

The difference, d, is called the common difference, and in general we say: d = Tn - Tn - 1 . An arithmetic sequence can also be called an Arithmetic Progression and the abbreviation AP is commonly used. Examples of arithmetic sequences a

An arithmetic sequence with 5 terms 4, +4

8,

12, +4

+4

16,

b

20

32, 27,

+4

-5

= T2 - T1 = T3 - T2

= 8-4 = 12 - 8

22, -5

17, -5

12, -5

7 -5

T1 = 32, T2 = 27, and so on.

T1 = 4, T2 = 8, and so on. d d

An arithmetic sequence with 6 terms

= 4 or = 4.

d d

The common difference is 4.

= T2 - T1 = T3 - T2

= 27 - 32 =-5 or = 22 - 27 =-5

The common difference is -5.

To find the common difference, find the difference between consecutive terms. Here is an example where the first term and common difference are used to find other terms. A sequence has T1 = 10 and a common difference of 5. Find T2 , T3 , T4 . Start with 10 and increase by 5 each term.

T1 = 10 T2 = 10 + 5 = 15 T3 = 15 + 5 = 20 T4 = 20 + 5 = 25

So the arithmetic sequence is: 10, 15, 20, 25. The common difference can be used to find missing terms. If the following sequence is arithmetic find the missing terms: 2, T2 , 14, 20, T5 , 32, 38 First, find d, by subtracting tow consecutive terms.

d = 20 - 14 = 6

Use T1 and d to find T2

` T2 = T1 + d = 2+6 =8

Use T4 and d to find T5

` T5 = T4 + d = 20 + 6 = 26

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Sequences and Series: Arithmetic 6. For each of these arithmetic sequences below, find (i) the number of terms in the sequence (iii) the fifth term, T5

7.

Questions

Basics

(ii) the first term, T1 (iv) the common difference, d

a

12, 16, 20, 24, 28, 32, ...

b

-10, -7, -4, -1, 2, 5, 8

c

1, 7, 13, 19, 25

d

-5, 7, 19, 31, 43, 55

e

0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30

f

4.2, 5.8, 7.4, 9.0, 10.6, 12.2

What are the next three terms if the common difference for these sequences is 5? a

b

3, 8, ...

-19, -14, ...

8. What are the next three terms if the common difference for these sequences is -4? a

b

17, ...

3, -1, ...

9. What is the second term if the common difference for these sequences is 7? a

5, T2 , 19 ...

b

-6, T2 , 8 ...

10. The fifth and seventh terms of an arithmetic sequence are 19 and 27 respectively. Find d.

11. The first term of an arithmetic sequence is 3 and d = -4. What is the seventh term of the sequence?

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Sequences and Series: Arithmetic

Basics

General Term of an Arithmetic Sequence The letter ‘a’ is used to represent the first term of a sequence. The letter ‘d’ is used to represent the common difference of an arithmetic sequence. For example The first term is a = 8

The common difference is d = 18 - 8 = 10

8, 18, 28, 38, ...

The next step is to work out a formula for any term of an arithmetic sequence. Here is an example. Finding the general term (also called the n th term) of an arithmetic sequence The sequence 3, 10, 17, 24, 31, 38, 45, 52 has a common difference of d = 7 T1 = 3 T2 = 10 = 3 + 7 T3 = 17 = 3 + 7 + 7 = 3 + 2^ 7 h T4 = 24 = 3 + 7 + 7 + 7 = 3 + 3^7h

Can you see the pattern? The general term is given by Tn = 3 + 7 ^n - 1h , and this simplifies to give Tn = 7n - 4 Starting from ‘a’ and adding the common difference ‘d’ for each term, we get: T1 = a

T2 = T1 + d

T3 = T2 + d

T4 = T3 + d

T5 = T4 + d

= a+d

= a + 2d

= a + 3d

= a + 4d

= a + ^2 - 1h d

= a + ^3 - 1h d

= a + ^4 - 1h d

= a + ^5 - 1h d

The formula for each term of an arithmetic sequence a, a + d, a + 2d, a + 3d, ... is: n th term

Tn = a + ^ n - 1h d a = first term

n = number of the term in the sequence

d = common difference

This is the General Term of an Arithmetic Sequence. Finding the general term of the following arithmetic sequence using the formula. The arithmetic sequence 2, 5, 8, 11, 14, 17, ... has first term a = 2 , and the common difference is d = 5 - 2 = 3 , so the formula for the n th term is Tn = a + ^ n - 1h d

= 2 + ^ n - 1h # 3 = 2 + 3n - 3

Tn = 3n - 1 The 10 th term is given by T10 = 3^10h - 1 = 29 .

The 15 th term is given by T15 = 3^15h - 1 = 44 .

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Sequences and Series: Arithmetic

Basics

Here is an example of using the formula to find ‘d’. Find d if a = 5 and T7 = 71 This is given:

a=5

n=7

T7 = 71

Substituting these into the formula Tn = a + ^ n - 1h d gives: 71 = 5 + ^7 - 1h d ` 71 = 5 + 6d ` d = 71 - 5 = 11 6 ` The common difference is d = 11

Sometimes a term that isn’t the first term is used to find terms of a sequence. In the next example the general term is needed to find the first term, T1 = a . The common difference of an arithmetic sequence is 14 and the twelfth term is 179. Find the first term. This is given:

d = 14

T12 = 179

Substituting these into the formula Tn = a + ^ n - 1h d gives:

179 = a + ^12 - 1h 14 a = 179 - ^11h 14 a = 25

So, the first term is a = 25 .

The next example has a negative d. The common difference of an arithmetic sequence is -8 and the twelfth term is -144. Find the first term. This is given:

d =-8

n = 16

T16 =-144

Substituting these into the formula Tn = a + ^ n - 1h d gives:

-144 = a + ^16 - 1h^-8h a =-144 + ^15h^8 h a=6

So, the first term is a = 6 .

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Sequences and Series: Arithmetic

Basics

Here is an example of finding the n th  term of a sequence. For the arithmetic sequence -12, -7, -2, 3, 8, 13, 18, ... a

Find a

b

a = T1 =-12 c

Find d d = T5 - T4 = 8 - 3 = 5

Find an expression for the n th  term Tn .

d

Find T20 T20 =-17 + 5^20h

Tn = a + ^ n - 1h d

=-12 + ^ n - 1h 5

= 83

` Tn =-17 + 5n

In the next example, two non-consecutive terms are used to find the general term of the sequence, Tn. The general term is used to form a system of two simultaneous equations which can be solved. In a sequence T8 = 149 and T12 = 225 . Find the general term. By substituting the information into the formula Tn = a + ^ n - 1h d two equations are formed, and these need to be solved simultaneously for a and d. Substituting gives us Tn = 149 , n = 8

T12 = 225 , n = 12

` 149 = a + ^8 - 1h d

and

` 225 = a + ^12 - 1h d

Solve these equations: 149 = a + 7d

1

225 = a + 11d

2

Subtracting 2 - 1 gives: 225 - 149 = 11d - 7d 76 = 4d ` d = 19 Substituting into 1 gives:

a = 149 - 7^19h ` a = 16 ` Tn = 16 + ^ n - 1h 19 ` Tn = 19n - 3

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Sequences and Series: Arithmetic

Questions

Basics

12. Find the formula for the n th term, Tn , of an arithmetic sequence with first term, a = 9, and common difference, d = 12.

13. For the arithmetic sequence 21, 16, 11, 6, 1, -4, -9: a

Find a

b

Find d

c

Find an expression for the n th term Tn .

d

Find T20

14. The fifth term of an arithmetic sequence is T5 = 17 . Using the formula Tn = a + ^ n - 1h d , and given that the first term is a = 1 , find the common difference d.

15. The first term of an arithmetic sequence is 21 and the tenth term equals -78. Find d.

16. A sequence has a common difference 9 and T20 = 174 . Find the first term.

17. An arithmetic sequence has T5 = 45 and T8 = 27 . Find the general term by substituting into Tn = a + ^ n - 1h d and solve a pair of simultaneous equations for a, d.

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Sequences and Series: Arithmetic

Knowing More

Series A series is the sum of the terms in a sequence. Comparing a sequence to a series. 5, 2, 13, 10, 8 is a sequence.

5 + 2 + 13 + 10 + 8 is a series.

Tn is used as a symbol for the n th term. Terms in a series are found the same way as terms in a sequence. Sn is the notation for the sum of the first n terms. Here are some basic examples. Consider the series 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + ... a

b

The second term of the sequence is T2 = 4 . The sum of the first two terms of the sequence is written S2 = T1 + T2 = 2 + 4 = 6 The fifth term is T5 = 10 . The sum of the first five terms is written S5 = T1 + T2 + T3 + T4 + T5 = 2 + 4 + 6 + 8 + 10 = 30

This example finds the sum to n terms for serveral n values. Given a series with 16 terms: 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53. Find S1, S4, S10, S16 S1 = T1 = 2 S4 = T1 + T2 + T3 + T4 = 2+3+5+7 = 17 S10 = T1 + T2 + T3 + T4 + T5 + T6 + T7 + T8 + T9 + T10 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 = 139 To find S16 the sum S10 can be used.

S16 = S10 + T11 + T12 + T13 + T14 + T15 + T16 = 129 + 31 + 37 + 41 + 43 + 47 + 53 = 381

The next example highlights the definition of the sum to n terms. A series has 16 terms and is given that S15 = 540 and S16 = 600 . Find the value of the 16th term, T16  . By definition,

S15 = T1 + T2 + ... + T15

S16 = T1 + T2 + ... + T15 + T16

Subtracting these two leaves us with just T16. S16 - S15 = ^T1 + T2 + ... + T15 + T16h - ^T1 + T2 + ... + T15h = T16 Therefore,

T16 = 600 - 540 = 60

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Sequences and Series: Arithmetic

Knowing More

Arithmetic Series Remember that in an arithmetic sequence, consecutive terms have the same difference. This holds in an arithmetic series also, consecutive terms always differ by the same amount, d, the common difference. The terms Tn + 1 and Tn are consecutive and for an arithmetic series: Tn = Tn + 1 + d = a + ^ n - 1h d

Tn - Tn + 1 = d Here are some arithmetic series. Examples of arithmetic series. a

b

S7 = 1 + 9 + 17 + 25 + 33 + 41 + 49 The common difference is d = 9 - 1 = 8.

S3 = 10 + 8 + 6 + 4 The common difference is d = 8 - 10 = -2.

Here are some basic examples to help understand arithmetic series Find the missing terms if the given series are arithmetic. a

b

13 + 30 + T3

9 + T2 + 31 T3 = 31 and a = 9

d = T2 - T1 = 30 - 13 = 17

` 9 + ^3 - 1h d = 31 ` 9 + 2d = 31

T3 = 30 + d = 30 + 17 = 47

Solve for d,

d = 31 - 9 = 11 2

So

T2 = 9 + 11 = 20

Here is an example of an arithmetic series with two unknown terms. Find the missing terms of the series 20 + T2 + T3 + 116 if it is an arithmetic series. Consecutive terms have a difference of d. T4 = 116

and

a = 20

` 20 + ^4 - 1h d = 116 20 + 3d = 116 d = 116 - 20 3 d = 32 So the missing terms are

T2 = 20 + 32 = 52

and

T3 = 52 + 32 = 84

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Sequences and Series: Arithmetic

Knowing More

Sum of an Arithmetic Series What about the general sum, Sn ? It is important to find a formula for Sn based on a, n, and d. Here is the derivation. The sum of the first n terms of an arithmetic series with first term, a, and common difference, d, is given by + ^a + d h

Sn = a In reverse:

+ ... + ^a + ^n - 2h d h +^a + ^n - 1h d h

+ ^a + ^n - 2 h d h

Sn = ^a + ^n - 1h d h

Adding these two rows gives:

+ ... + ^a + d h

+a

` Sn + Sn = 6a + a + ^ n - 1h d @ + 6^a + d h + a + ^ n - 2h d @ + ... + 6a + ^ n - 2h d + ^a + d h@ + 6a + ^ n - 1h d + d @ ` 2Sn = 62a + ^ n - 1h d @ +

62a + ^ n - 1h d @

1

+ ... + ...

2

There are n lots of ^2a + ^n - 1h d h Finally dividing by 2 gives the general term. The first term a

62a + ^ n - 1h d @

n-1

n

` 2Sn = n 62a + ^n - 1h d@

Sn = n 62a + ^ n - 1h d @ 2

Sum to n terms:

+ 62a + ^ n - 1h d @

The number of terms, n

The common difference d

This is important if you want to find S100 or S200 . Here is an example. Find the sum of the first 25 terms of the arithmetic series 4 + 10 + 16 + 22 + 28 + 34 + ... a = T1 = 4,

d = 10 - 4 = 6

n = 25

S25 = n 62a + ^ n - 1h d @ 2 = 25 62^4h + ^25 - 1h 6 @ 2 = 1900

The sum to 25 terms is 1900.

The next example uses the sum of two terms to find a and d, which are used in the sum formula. The third term of an arithmetic series is given by T3 = 56 and the sum of the 3rd and 4th terms is T3 + T4 = 140 . a

Find the fourth term, T4

Subtracting gives

b

Find the sum to 40 terms, S40 .

T3 = 56

d = T4 - T3 = 84 - 56 = 28

T3 + T4 = 140

a = T1 = T2 - d = T3 - 2d

T4 = 140 - 56 = 84

` a = 56 - 2^28h = 0

S40 = n 62a + ^ n - 1h d @ 2 = 40 62^0h + ^40 - 1h 28 @ 2 S40 = 21840

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Sequences and Series: Arithmetic

Knowing More

If the last term of the series l is known then the formula Sn = n 62a + ^ n - 1h d @ becomes 2 n Sn = ^ a + l h 2

The last term is l = a + ^ n - 1h d

The first term a

Sometimes n needs to be found before being able to calculate Sn. Use the formula for the last term Tn to find the number of terms, n. Find the sum of the arithmetic series 5 + 10 + 15 + 20 + 25 + ... + 2555 . a=5

d=5

l = 2555

Step 2: Substitute known values ` S511 = n ^a + lh 2 = 511 ^5 + 2555h 2

Step 1: Find n. Tn = a + ^n - 1h d = 5 + ^n - 1 h 5 = 5n Tn = 2555

= 654080

` 5n = 2555 n = 2555 = 511 5 Here is an example where the value of n remains unspecified. The sum of the first n even numbers is an arithmetic series and Sn = 2 + 4 + 6 + 8 + ... + 2n a

What is the general term, Tn?

b

The last term in the series sum gives away the general term, ` Tn = 2n . Check that the formula gives 2, 4, 6, 8 … when substituting = 1, 2, 3, 4 … .

Use the formula Sn = n ^a + l h to find the 2 formula for the sum to n terms, Sn. a = 2, l = 2n and since there are n terms the sum is Sn = n ^a + lh 2 = n ^2 + 2nh 2 = 2n ^n + 1h 2 = n ^n + 1 h

An arithmetic series has sum Sn = 5 n ^ n + 1 h . How many terms must be taken for the sum to exceed 200? 6 Use trial and error for different values of n. When n = 5, S5 = 5 ^5 h^6 h = 25 . When n = 10, Sn = 5 ^10h^11h = 275 . 6 6 3 5 When n = 15, S15 = ^15h^16h = 200 . This is still not greater than 200 so we increase n by 1 again.. 6 5 When n = 16, S16 = ^16h^17h = 680 2 200 6 3 ` 16 terms must be taken so the sum exceeds 200.

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Sequences and Series: Arithmetic

Questions

Knowing More

18. If these are arithmetic sequences find the value of the missing terms a

6 + 25 + T3

b

c

20 + T2 + 4

12 + T2 + T3 + 39

19. Find the sum to 25 terms of the arithmetic series using the formula Sn = n 62a + ^ n - 1h d @. 2 a 32 + 30 + 28 + 26 + 24 + 22 + 20 + 18 + 16 + 14 + 12 + ...

b

- 25 - 21 - 17 - 13 - 9 - 5 - 1 + 3 + 7 + ...

c

5 + 3 + 1 - 1 - 3 - 5 - 7 - 9 - 11 - 13 - 15 - 17 - ...

20. Find the sum of the series using the first and last term formula, Sn = n ^a + lh . 2 a 3 + 7 + 11 + 15 + 19 + 23 + 27 + 31 + 35 + 39 + 43

b

- 9 - 16 - 23 - 30 - 37 - 44 - 51 - 58 - 65 - 72

c

- 1 - 10 - 19 - 28 - 37 - 45 - 54 - 63 - 72 - 81 - 90 - 99

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Sequences and Series: Arithmetic

Questions

Knowing More

21. Find the sum of these series. a

2 + 4 + 6 + 8 + 10 + ... + 200.

b

93 + 87 + 81 + ... + 6 + 3

c

-1 + 0 + 1 + 2 + 3 + 4 + ... + 199

d

250 + 238 + ... - 134

22. An arithmetic series has 20 terms with first and last terms 5 and 195 respectively. Find the sum of the series.

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Sequences and Series: Arithmetic

Questions

Knowing More

23. A series has T3 = 4 and T2 + T3 =-12 . Find T2 .

24. The fourth term of an arithmetic series is 28 and the sum of the fourth and fifth terms is 32. a

Find the fifth term.

b

Find the common difference, d.

c

Using the formula for the nth term, Tn = a + ^ n - 1h d with n = 5 and your answers from above, to find the value of a.

25. For the arithmetic series 1744 + ... + 28 + 22 + 16 + 10 + 4 a

Find the values of a and d and write down the nth term Tn .

b

By using the last term, 4, and the formula for Tn , find out how many terms are there in the series?

c

Find the sum of this series.

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Sequences and Series: Arithmetic

Questions

Knowing More

26. a Find the sum of the first 5 multiples of 3, that is, 3 + 6 + 9 + 12 + 15 .

18

b

Is this an arithmetic sequence? If so, find d.

c

The sum of the first n multiples of 3 is an arithmetic series and Sn = 3 + 6 + 9 + 12 + ... + 3n Find the general term Tn .

d

Use the formula Sn = n ^a + lh to find the sum to n terms and show this sum is Sn = 3 n^ n + 1h 2 2

e

Find n if the sum is 315.

f

How many terms must be taken for the sum to exceed 560? (Use trial and error approach)

g

Why can this sum never be 170?

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Sequences and Series: Arithmetic

Using Our Knowledge

Applications of Arithmetic Series Arithmetic series have uses in the real world. The next example is an application of arithmetic series to a financial problem involving annual salary increases. The starting salary of an accountant is $34 000 per year. After each year of employment he will receive an increase in the salary by an amount $1200 . a

What is his salary in the ninth year of employment? Salary for first year

= $34 000

Salary for second year is: $34 000 + $1200 Salary for third year is:

$34 000 + $1200 + $1200 = $34000 + $1200 # 2

` Salary for ninth year is $34000 + $1200 # 8 Or using the formula for arithmetic series with: a = $34000

d = $1200

n=9

Tn = a + ^ n - 1h d

Tn = $34000 + $1200^ n - 1h Therefore,

a

T9 = $34000 + $1200^9 - 1h = $43600

This is the last term in the series

How much are his total earnings for the first nine years. The total salary in the first nine years is the sum to 9 terms of an arithmetic series. Denote the total earnings to the end of the nth year by Sn (sum to n terms). Since we know n = 9 and the last term is l = 43600 then the formula for the sum to n terms of an arithmetic series gives: Sn = n ^a + lh 2 = 9 ^34000 + 43600h 2 = 349200 Total earnings are $349 200

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Using Our Knowledge

Here is another real world problem. A section of a stadium has 7 seats in the first row, 10 seats in the second row, 13 seats in the third row, and so on. If the last row has 91 seats, how many seats are in this section?

First row, 7 seats

Second row, 10 seats

Third row, 13 seats

. . . .

.

.

Last row, 91 seats

This can be expressed as an arithmetic sum as: 7 + 10 + 13 + ... + 91 a=7 d=3 To find the sum, the number of terms, n, is needed. The nth term of this series is given by: Tn = a + ^ n - 1h d

= 7 + ^ n - 1h 3 = 3n + 4

The last term is 91. Which term is this? Solve: Tn = 91 3n + 4 = 91 3n = 87 n = 29 So there are 29 terms. The sum of the arithmetic series using the formula is: S29 = n ^a + lh 2 = 29 ^7 + 91h 2 S29 = 1421 So there are 1421 seats in the section.

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Using Our Knowledge

Here is another more complicated real world problem. A groundskeeper distributes fertilizer on the green using a wheelbarrow. Over several days he empties the wheelbarrow in 20 metre increments, and has to go back to refill each time. If he empties 32 barrowfulls, how far does he walk? Emptying wheelbarrow

20 m

40 m

640 m

1

2

32

Going back to refill the wheelbarrow

For the first load he walks 20 m and returns, so he has travelled 2 # 1 # 20 m . For the second load he walks 2 # 20 m and returns, so he has travelled 2 # 2 # 20 m and so on. In total he will travel 2 # 1 # 20 + 2 # 2 # 20 + 2 # 3 # 20 + ... + 2 # 32 # 20 = 40 + 80 + 120 + 160 + ... + 1280 This is an arithmetic series. a = 40 d = 40 n = 32 l = 1280 (last term) The sum of the arithmetic series is: S32 = n ^a + lh 2 32 ^40 + 1280h = 2 = 21120 So the groundskeeper walks a total of 21 120 metres.

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Using Our Knowledge

Here is an example relating to building. A floor in room has the shape of a trapezium as shown. It is to have floor boards put in. The difference between the lengths of adjacent boards is a constant and so the lengths of the boards form an arithmetic sequence. The shortest board is 4 m in length and the longest board is 8 m. The sum of the lengths of the boards is 270 m. a

Find the number of boards. a = 4, l = 8 Sn = n ^a + lh 2 270 = n ^4 + 8h 2 2^270h n= 12

Floor

4 m

8 m

n = 45

b

Find the difference in length between adjacent boards in centimetres (correct to 2 decimal places) The difference in length between adjacent boards is the common difference d.

The last board is number 45.

Tn = a + ^n - 1h d

` 4 + ^45 - 1h d = 8 d = 1 = 0.091 m 11 = 9.09 cm (2 d.p.)

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Questions

Using Our Knowledge

27. The annual salary of a sales manager increases by $750 each year, where in his first year he earned $45 000. a

What is his salary in the sixth year of employment.

b

How much money did he earn in total in the first 5 years of employment?

28. Chairs in an amphitheatre are arranged in an increasing order where the number of chairs in a row is increasing from 6 until 58. Find the number of chairs if the difference in the number of chairs in rows next to each other is 2.

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Questions

Using Our Knowledge

29. An architect is finding the cost of building a multi storey building. The 1st floor costs $200 000, the 2nd floor costs $225 000 and each subsequent floor costs $25 000 more to build than the floor below. What does it cost to build a 40-storey building?

30. The temperature in a high pressured hot water tank is falling at a constant rate. A reading was taken each 10 minutes and these were: 230c, 224c, 218c, ... The final reading taken was equal to 26c . How many readings were taken altogether.

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Sequences and Series: Arithmetic

Thinking More

Sigma Notation The Greek sigma (/ ) symbol is used to write series easily. / is a short hand notation for describing series in a precise and concise way. Writing a sum in sigma notation relies on noticing the pattern in the series. In sigma notation, we have The last value of n is n = 5 5

/ Tn = T1 + T2 + T3 + T4 + T5

n=1

The general expression for the n th term is Tn

The first value of n is n = 1

Write out the terms of the series given in sigma notation as

8

/ 2n .

n=1

End 8

/ 2n

Formula

n=1

Start

The bottom number, ‘n = 1 ’, tells you to start the first term by substituting in n = 1 . Then substitute n = 2 and add it on, then n = 3 and add it on again, all the way up to n = 8 . Expanding the sigma notation one gets n=1

n=2

n=3

n=6

n=8

/ 2n = 2^1 h + 2^2h + 2^3h + 2^4h + 2^5h + 2^6h + 2^7h + 2^8h 8

n=1

= 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16

Here is another example. Write out the series in expanded form

6

/ 4n + 1 .

n=0

/ 4n + 1 = ^4^0h + 1h + ^4^1 h + 1h + ^4^2h + 1h + ^4^3h + 1h + ^4^4h + 1h + ^4^5h + 1h + ^4^6h + 1h 6

n=0

= 1 + 5 + 9 + 13 + 17 + 21 + 25

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Thinking More

Here is an example with negative numbers where more care is needed. Write out the series in expanded form

2

/ 4 - n.

n =-3

/ 4 - n = ^4 - ^-3hh + ^4 - ^-2hh + ^4 - ^-1hh + ^4 - 0h + ^4 - 1h + ^4 - 2h 2

n =-3

= ^4 + 3h + ^4 + 2h + ^4 + 1h + 4 + 3 + 2 = 7+6+5+4+3+2

Here a two examples of writing the expanded form from sigma notation. Write out in expanded form (without sigma notation). a

6

/ 5n + 4

Start at n = 2 and end at n = 6

n=2

= ^5 ^2h + 4h + ^5 ^3 h + 4h + ^5 ^4h + 4h + ^5 ^5 h + 4h + ^5 ^6 h + 4h = 14 + 19 + 24 + 29 + 34 = 120 5

b

/ -5n - 7

Start at n = 3 and end at n = 5

n=3

= 6-5 ^3 h - 7@ + 6-5 ^4h - 7@ + 6-5 ^5 h - 7@ =-15 - 7 - 20 - 7 - 25 - 7 =-81

Some examples of writing a series in the sigma notation follow. Write the series in sigma notation: 5 + 11 + 17 + 23 + 29 + 35 + 41 + 47 This is an arithmetic series with common difference, d = 11 - 5 = 6 , and a = 5 . ` Tn = 5 + 6^ n - 1h = 6n - 1 Therefore the sum is

8

/ 6n - 1

n=1

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Thinking More

The next example involves recognising that the series is arithmetic and finding the sum. Calculate the sum of the series:

100

/ 7m + 4

m=0

Write it out in expanded format 100

/ 7m + 4 = 4 + 11 + 18 + ... + 704

m=0

This is an arithmetic series, a = 4,

d = 7,

l = 704,

n = 101 (number of terms)

Sn = n ^2a + d^ n - 1hh 2 S101 = 101 ^2 # 4 + 7^101 - 1hh 2 = 35754

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Questions

31. Write out in expanded form (without sigma notation) and evaluate a

5

/ 7j

b

8

/ 4 - 9k

k=1

j=0

32. Write these series in sigma notation a

b

6 + 12 + 18 + 24 + 30 + 36

2 + 9 + 16 + 23 + 30

33. Calculate the sum of each of these series. a

7

/ 9j + 5

b

j=1

28

233

/ 4 - 6n

n = 90

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Thinking More

Sequences and Series: Arithmetic

More Questions

Basics 1. What is the general term of the sequence 6, 8, 10, 12, ...? a

Tn = 2 + 4n

b

c

Tn = 4 + 2n

Tn = 6n + 2

d

Tn = 6 + 8n

d

67

2. What is the 3rd term of the sequence with general term Tn = 4n + 3 ? a

7

b

c

12

15

3. If the numbers 12, x, 24 are in arithmetic progression, what is the value of x? a

15

b

16

c

18

d

20

c

7

d

8

4. If Tn = 19 - 9n , what is the value of T3? a

-8

b

-7

5. Which of the following does not represent an arithmetic series? a

b

c

d

6. For the sequences shown, write down the number of terms, and the values of T1, T3, T6. a

b

3, 6, 1, -7, 0, 45

9, 6, 2, -3, -10, 20, 4

7. Write down the first 5 terms of the sequence whose general term is given a

Tn = 4n

b

Tn = 3 + 5n

c

Tn = 9 - n

d

b

Tn =-3 ^4 - 5nh

Tn =-7n - 14

8. What is the 5th term of the sequence with nth term

9.

a

Tn = 11 - 7n

a

The nth term of a sequence is given by Tn = 8 - 5n . For which value of n does the nth term equal -162?

b

The nth term of a sequence is given by Tn = 11n + 29 . For which value of n does the nth term equal 260?

10. Find the general term, Tn , of these sequence 1 , 2 , 3 , 4 , ... a 2, 22, 42, ... b 7 7 7 7

c

-6, -1, 4, 9, 14, 19, ...

11. What are the next three terms if the common difference is 7?

a

7, ...

b

-19, ...

12. What are the next three terms if the common difference is -8?

a

34, ...

b

-19, ...

13. What is the missing term if the common difference is 8?

a

5, T2, 21, ...

b

-16, -8, T3, ...

14. The fourth and eleventh terms of an arithmetic sequence are 69 and 167. Find the common difference. 15. The first term of an arithmetic sequence is 19 and the common difference is -9. What is the seventh term of the sequence? 16. Find the formula for the nth term, Tn, of an arithmetic sequence whose first term, a, and common difference, d, are given by a = 13 and d = -8.

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More Questions

17. For the arithmetic sequence 12, 25, 38, 51, ... a

Find a, d

b

Find the expression for the nth term, Tn.

c

Find T20

18. The fifteenth term of an arithmetic sequence is T15 = 20 . Using the formula Tn = a + ^n - 1h d , and given that the first term is a = 188, find the common difference d. 19. The first term of an arithmetic sequence is -26 and the eleventh term equals 64. Find the common difference. 20. The common difference of an arithmetic sequence is 17 and the twentieth term equals 347. Find the first term. 21. The sixth and tenth terms of an arithmetic sequence are given by T6 =-7 and T10 =-27 . Find the general term by substituting into Tn = a + ^n - 1h d , and solve a pair of simultaneous equations for a, d. 22. Is 35 a term of the sequence with general term Tn = 76 - 4n ? 23. Which term of the sequence 5, 13, 21, 29, 37, 45, ... is 341? 24. Write down the numbers 18 and 46 and insert three numbers between them so as to give 5 numbers in arithmetic progression. 25. The sum of the first two terms of an arithmetic sequence is 10 and the sum of the next two terms is 18. Find the first term and common difference. 26. If the sequence 1 , 1 , 1 is arithmetic, show that 2ac = b^a + ch . a b c 27. Find x and the common difference if the following are AP’s a

b

20, x, 30

x + 2, 10, x + 8

c

5x - 2, x + 5, 3x

28. Find the common difference of the AP, then find x, given that T10 = 25 ;

7x - 12, 7x - 2, 7x + 8

Knowing More 29. If the sequence 8, T2, T3, 29 is arithmetic, what are the missing terms? a

b

13, 20

13.25, 21.125

30. What is the sum of the series -4 - 3 - ... + 42 ? a S47 = 47 ^-4 + 42h b S46 = 46 ^-4 + 42h 2 2

c

15, 22

S45 = 45 ^-4 + 42h 2 31. How many terms are there in the arithmetic series 5 + 7 + 9 + 11 + ... + 31 ? a

b

13

c

c

14

19

d

16, 24

d

S47 =- 46 ^-4 + 42h 2

d

20

32. The third term of an arithmetic series is given by T3 = 28 and the sum of T3 + T4 =- 56 is given. What is the fourth term T4 ? a

b

-84

-28

33. If the sums S9 = 65 and S8 = 92 , what is T9 ? a -27 b 92 65

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c

28

d

84

c

27

d

157

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34. If these are arithmetic series find the value of the missing terms a

23 + 50 + T3

b

-25 + T2 + 1

c

6 + T2 + T3 - 9

35. Find the sum to 25 terms of the arithmetic series using the formula Sn = n ^2a + ^ n - 1h d h . 2 a 62 + 60 + 58 + 56 + 54 + 52 + 50 + ... b

-90 - 81 - 72 - 63 + ...

c

-1 - 2 - 3 - 4 -5 - 6 - 7 + ...

36. Find the sum of the series using the first and last term formula, Sn = n ^a + lh 2 a 1 + 2 + 3 + 4 + ... + 1000 b

-4 - 9 - 14 - 19 - 24 - 29 - 34 - 39 - 44 - 49 - 54 - 59 - 64 - 69 - 74 - 79

c

50 + 29 + 8 - 13 - 34 - 55 - 76 - 97 - 118 - 139 - 160 - 181 - 202 - 223 - 244 - 265

37. Find the sum of these series by first finding the number of terms, n, in the series using the last term and the formula Tn = a + ^ n - 1h d a

3 + 6 + 9 + 12 + ... + 1098

b

-4 - 8 - 12 - 16 - 20 - ... - 516

c

-23 - 16 - 9 - 2 + ... + 110

d

-1022 - 1000 - ... - 142

38. An arithmetic series has 41 terms with first and last terms -100 and 140 respectively. Find the sum of the series. 39. The seventh term of an arithmetic series is given by T7 = 29 and the sum of T7 + T8 =-905 is given. What is the eight term T8 ? 40. The fifth term of an arithmetic series is -1 and the sum of the fourth and fifth terms is -15. a

Find the fourth term.

c

Using the formula for the nth term, Tn = a + ^ n - 1h d and your answers from above, to find the value of a.

b

Find the common difference, d.

41. For the arithmetic series 13 + 17 + 21 + ... + 6613. a

Find the values of a and d and write down the nth term Tn.

b

By using the last term, 6613, and the formula for Tn, find out how many terms are there in the series?

c

Find the sum of this series.

42. a Find the sum of the first 9 multiples of 6, that is, 6 + 12 + 18 + 24 + 30 + 36 + 42 + 48 + 54. b

Is this an arithmetic sequence? If so, find d.

The sum of the first n multiples of 6 is an arithmetic series and Sn = 6 + 12 + 18 + 24 + ... + 6n. Find the general term Tn. d Use the formula Sn = n ^ a + lh to find the sum to n terms and show this sum is Sn = 3n^ n + 1h 2 e Find n if the sum is 2610. c

f

How many terms must be taken for the sum to exceed 7650? (Use trial and error approach, or logs)

g

Give a simple reason why can this sum never be 695?

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43. Find the sum of all positive multiples of 9 which are less than 230. 44. a Show that the sum to n terms of the AP 1 + 3 + 5 + ... is Sn = n2 b

Find the sum of the first 100 odd numbers.

45. Find the sum of the integers between between 150 and 250 which are multiples of 7. 46. The sum of the first 5 terms of an arithmetic series is -10, and the sum of the next two terms is -32. Find a, d. 47. Find the sum of the AP: x + 2x + 3x + ... + nx . 48. Find the sum: 18 + 50 + 98 + ... + 45 2

Using Our Knowledge 49. The Chairs in a small amphitheatre are such that the first row has 9 chairs, and each row increases by 4 until the last row, which has 33 chairs. What is the total number of chairs. a

b

37

c

114

138

d

147

50. Three numbers x, y, z are inserted between 0 and 256, so as to give 5 numbers in AP. What are the values of x, y, z? a

b

4, 16, 64

16, 64, 128

c

32, 64, 128

d

64, 128, 192

51. A man earns $30 000 in his first year at work, and gets an increase in his salary by $1250 each year. What is his salary in the 8th year of his employment? a

b

$38 750

c

$39 250

$40 000

d

$41 250

d

1, 34

d

2 + 4 + ... + 40

52. The sequence x, 7, 17, y are in arithmetic progression. What are the numbers x, y? a

b

3, 27

c

-3, 27

1, 24

53. Which of the following is an arithmetic series with 21 terms whose sum is 504? a

2 + 3 + ... + 40

b

4 + 6 + ... + 44

c

-5 - 4 - ... + 26

54. The annual salary of a tradesman increases by $1250 each year, where in his first year he earned $32 000. a

What is his salary in the fourth year of employment?

b

How much money did he earn in total in the first 7 years of employment?

55. Chairs in an amphitheatre are such that the first row has 12 chairs, the second row has 16 chairs, and each row increases by 4, until the last row, which has 104 chairs. Find the total number of chairs. 56. An architect is finding the cost of building a multi storey car park. The first floor costs $170 000, the second floor costs $190 000, and $60 000 for each additional floor. What is the cost of building a 12 storey car park? 57. A new building has 26 floors. The cost of building each floor varies. The first floor costs $2 000 000. The cost of building each subsequent floor will be $650 000 more than the floor immediately below.

32

a

What will be the cost of building the 9th floor?

b

What will be the cost of building all 26 floors?

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58. Paula is training for a 4 km swimming race by swimming each week for 30 weeks. She swims 200 m in the first week, and each week after that she swims 200 m more than the previous week, until she reaches 4 km in a week. She then continues to swim 4 km each week. a

How far does Paula swim in the fourth week?

b

In which week does she first swim 4 km?

c

What is the total distance Paula swims in 30 weeks?

59. The temperature in a cool room was taken at regular intervals after it was turned on, and the readings in degrees Celsius were 25c, 24.1c, 23.2c, ... . Assume that these readings are in arithmetic progression. If the final reading taken was equal to -9.2c , how many readings were taken altogether? 60. A tall fence has the shape of a trapezium and has planks arranged as shown. The difference between the lengths of adjacent planks is a constant and so the lengths of the planks form an arithmetic sequence. The shortest plank is 180 cm in length and the longest string is 250 cm. The sum of the lengths of the planks is 774 m a

Find the number of planks.

b

Find the difference in length between adjacent planks.

Thinking More 61. What is the expanded form of the series

6

/ 9m ?

m=1

a

9 + 18 + ... + 54

1 + 9 + ... + 54

b

62. What is the expanded form of the series

c

9 + 18 + ... + 56

d

1 + 18 + ... + 54

c

-3 + 0 + 3 + 6

d

0 + 9 + 24

/ 3k + 4

d

/ 3k - 2

S10 = 10 ^12 + 39h 2

d

S9 = 9 ^12 + 36h 2

b

/ 3j 3

3

/ 3k 2 - 3 ?

k=0

a

3 - 1 + 9 + 24

b

-3 + 0 + 9 + 24

63. What is the sigma notation for the series 4 + 7 + 10 + 13 + 16 + 19? a

6

/ 4n + 3

7

/ 3k - 2

b

n=1

c

k=2

64. What is the sum of the series

7

7

n=2

0

12

/ 3k + 3 ?

k=3

a

S9 = 9 ^12 + 39h 2

b

S10 = 10 ^9 + 36h 2

c

65. Write out in expanded form (without sigma notation) and evaluate

6

a

/ 4j j=1

5

j=0

66. Write these series in sigma notation. a

7 + 14 + 21 + 28 + ... + 70

b

-30 - 20 - 10 - 0 + 10 + 20 + 30 + 40

c

1000 + 999 + 998 + 997 + 996 + 995+ ... + 1

67. Calculate the sum of each of these series.

a

10

/ 5j - 9 j=4

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100

/1 - n

n=0

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Sequences and Series

Answers

Basics 1.

2.

c

a

The number of terms in the sequence is 6 T1 = 3 T2 = 6 T3 = 1 T4 =-7

b

The number of terms in the sequence is 7 T1 = 9 T2 = 6 T3 = 2 T4 =-3

T3 = 30

Knowing More 19. a S25 = 900 c

20. a S11 = 253

7, 14, 21, 28, 35

b

13, 19, 25, 31, 37

c

-2, -5, -8, -11, -14

d

-19, -38, -57, -76, -95

3.

a

T5 = 140

b

T5 =-55

22. S20 = 2000

4.

a

n = 21

b

n = 15

23. T2 =-8

5.

a

Tn = 2n

b

Tn = 3n + 1

24. a T5 = 4

6.

a

(i) 6 (ii) T1 = 12 (iii) T5 = 28 (iv) d = 4

b

(i) 7 (ii) T1 =-10 (iii) T5 = 2 (iv) d = 3

(i) 5 (ii) T1 = 1 (iii) T5 = 25 (iv) d = 6

d

(i) 11 (ii) T1 = 0 (iii) T5 = 12 (iv) d = 3

f

c

e

7. 8. 9.

a

T3 = 13

b

T3 =-9

a

T2 = 13

T3 = 9

b

T3 =-5

T4 =-9

a

T2 = 12

T4 = 18 T4 =-4

(i) 6 (ii) T1 =-5 (iii) T5 = 43 (iv) d = 12 (i) 6 (ii) T1 = 4.2 (iii) T5 = 10.6 (iv) d = 1.6

T5 = 23

21. a

S100 = 10 100

b

S16 = 768

c

S201 = 19 899

d

S33 = 1914

b

d =-24

c

25. a a = 1735, d =-6, Tn =-6n + 1750 b

c

S291 = 254 334

26. a 45

b

Yes. d = 3

c

Tn = 3n

e

n = 14

f

19

g

170 is not a multiple of 3

27. a T6 = $48 750

d =-5

32. a

T20 =-74

33.

15. d =-11 17. Tn = 6n - 3

34

6

/ 6n

b

a

S7 = 287

b

More Questions

16. a = 3

b

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S5 = $232 500

-5 - 14 - 23 - 32 - 41 - 50 - 59 - 68 =-292

n=1

14. d = 4

18. a T3 = 44

b

31. a 7^0h + 7^1 h + 7^2h + 7^3 h + 7^4h + 7^5 h = 105 b

d

n = 291

Using Our Knowledge

12. Tn = 12n - 3 Tn =-5n + 26

a = 100

Thinking More

T2 = 1

11. T7 =-21

c

S13 =-600

30. n = 35

T5 =-13

b

S10 =-405

29. S40 = $27  500 000

T4 = 5

10. d = 4

13. a a = 21

b

28. S27 = 864 chairs

T5 = 1

b

S25 = 575

S25 =-425

a

c

b

T2 = 12

1.

b

Tn = 4 + 2n

2.

c

15

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/ 7n - 5

n=1

S7 =-155 664

Sequences and Series

Answers

3.

c

18

32. a -84

4.

a

-8

33. a -27

5.

b

34. a T3 = 77 c

6.

a

6 terms. T1 = 3, T3 = 1, T6 = 45

35. a

b

7 terms. T1 = 9, T3 = 2, T6 = 20

c

a

4, 8, 12, 16, 20

c

8, 7, 6, 5, 4

d

-21, -28, -35, -42, -49

8.

a

T5 =-24

b

9.

a

n = 34

b

7.

10. a Tn = 20n - 18 c

b

8, 13, 18, 23, 28

36. a c

37. a c

T5 = 63

b

T2 =-12

b

S25 = 450

b

S16 =-664

Sn = 201 483

b

Sn =-33 540

Sn = 870

d

Sn =-23 862

b

d = 13

T2 = 1, T3 =-4 S25 = 950 S25 =-325 S1000 = 500 500 S16 =-1720

38. S41 = 820

n = 21 b Tn = n 7

39. T8 =-934 40. a T4 =-14

Tn = 5n - 11

11. a 14, 21, 28

b

-12, -5, 2

12. a 26, 18, 10

b

-27, -35, -43

13. a T2 = 13

b

T3 = 0

c

14. d = 14

a =-53

41. a a = 12, d = 13, Tn = 4n + 9 b

n = 1651

c

S1651 = 5 459 763

42. a

S9 = 270

b

Yes, d = 6

15. T7 =-35

c

Tn = 6n

e

n = 29

16. Tn = 21 - 8n

f

n = 51

g

695 is not a multiple of 6

b

S7 = $250 250

b

S26 = $263 250 000

17. a a = 12, d = 13 c

b

Tn = 13n - 1

T20 = 259

44. b S100 = 10 000

18. d = -12

45. S14 = 2793

19. d = 9

46. a = 6, d = -4 nx^ n + 1h 47. Sn = 2 48. S22 = 528 2

20. a = 24 21. Tn = 23 - 5n 22. No, 35 is not a term in the sequence.

49. d 147

23. n = 43

50. d 64, 128, 192

24. 18, 25, 32, 39, 46

51. a $38 570

25. a = 4, d = 2 27. a d = 5, x = 25 c

28. 29. 30. 31.

43. S25 = 2925

b

52. b -3, 27

d = 3, x = 5

53. b 4 + 6 + ... + 44

d = -1, x = 2

54. a $35 750

d = 10, x = - 53 7 c 15, 22 a S47 = 47 ^-4 + 42h 2 b 14

55. S24 = 1392 chairs 56. $960 000 57. a T9 = $7 200 000

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L

S1

SERIES

TOPIC NUMBER

35

Sequences and Series 58. a 800 m. c

Answers

b

20th week.

b

d = 2 cm

82 km

59. 39 readings 60. a 36 planks 61. a 9 + 18 + ... + 54 62. b -3 + 0 + 9 + 24 63. b

7

/ 3k - 2

k=2

64. c

S10 = 10 ^12 + 39h 2

65. a

/ 4j = 4 + 8 + 12 + 16 + 20 + 24 = 84

6

j=1

b

5

/ 3j3 = 0 + 3 + 24 + 81 + 192 + 375 = 675 j=0

66. a

10

/ 7k

b

k=1

c

/ ^10k - 40h 8

k=1

999

/ 1000 - k

k=0

67. a S7 = 182

36

b

L

S1

SERIES

TOPIC NUMBER

S101 =-4949

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Sequences and Series: Arithmetic

Notes

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L

S1

SERIES

TOPIC NUMBER

37

Sequences and Series: Arithmetic

38

L

S1

SERIES

TOPIC NUMBER

Notes

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Sequence & Series - Arithmetic

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