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An Arithmetic Sequence is a sequence where each term

is formed from the preceding term by adding a constant

to the preceding term.

Consider the sequence -3, 1, 5, 9.

This sequence is found by adding 4 to the previous term.

The constant term which is added to each term to

produce the sequence is called theCommon Difference.

2.5.2

9

1

5

-3

-3

-3 + 4

-3 + 4 + 4

-3 + (2)4

-3 + 4 + 4 + 4

-3 + (3)4

-3 + (1)4

a + d

a + 2d

a

a + 3d

Continuing with this pattern, the general term is derived as:

tn = a + (n - 1) d

2.5.3

The General Arithmetic Sequence

General

Term

Number or

Position of

the Term

tn = a + (n - 1) d

First

Term

Common

Difference

2.5.4

Finding the Terms of an Arithmetic Sequence

Given the sequence -5, -1, 3, …:

a) Find the common difference.

d = t2 - t1

= (-1) - ( -5)

= 4

Note: The common difference

may be found by subtracting

any two consecutive terms.

c) Find the general term .

b) Find t10 .

tn = a + (n - 1) d

a = -5

n = ?

d = 4

tn = a + (n - 1) d

a = -5

n = 10

d = 4

tn = ?

= -5 + (n - 1) 4

= -5 + 4n - 4

tn = 4n - 9

t10 = -5 + (10 - 1) 4

= -5 + (9) 4

t10 = 31

a = -5

n = ?

d = 4

tn =63

d) Which term is equal to 63?

63 = - 5 + 4n - 4

72 = 4n

18 = n

tn = a + (n - 1) d

63 = -5 + (n - 1) 4

t18 = 63

2.5.5

Finding the Number of Terms of an Arithmetic Sequence

Find the number of terms in 7, 3, -1, - 5 …, -117 .

tn = a + (n - 1) d

a = 7

n = ?

d = -4

tn =- 117

-117 = 7 + (n - 1) (-4)

-117 = 7 - 4n + 4

-117 = -4n + 11

-128 = -4n

32 = n

There are 32 terms in

the sequence.

A pile of bricks is arranged in rows. The number of bricks

in each row forms a sequence 65, 59, 53, …, 5.

Which row contains 11 bricks? How many rows are there?

tn = a + (n - 1) d

tn = a + (n - 1) d

a = 65

n = ?

d = - 6

tn =5

a = 65

n = ?

d = - 6

tn =11

5 = 65 + (n - 1) (-6)

-66 = -6n

n = 11

11 = 65 + (n - 1) (-6)

-60 = -6n

10 = n

The 10th row contains 11 bricks.

There are 11 rows in this pile.

2.5.6

Arithmetic meansare the terms that are between

two given terms of an arithmetic sequence.

Insert five arithmetic means between 6 and 30.

6 _ _ _ _ _ 30

7 terms altogether

tn = a + (n - 1)d

a = 6

n = 7

d = ?

tn =30

30 = 6 + (7 - 1)d

30 = 6 + 6d

24 = 6d

4 = d

Therefore, the terms are:

26

6, , 30

10,

14,

18,

22,

2.5.7

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