MATHPOWERTM 10, WESTERN EDITION

Chapter 2 Number Patterns2.5

2.5.1

An Arithmetic Sequence is a sequence where each termis formed from the preceding term by adding a constantto 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 the Common Difference.

2.5.2

Arithmetic Sequences

t 1 t 2 t 3 t 4

Sequence

Sequence expressed using the common

difference

Sequence expressed in general

terms

-3 + (1)4

-3 + 4 + 4

-3 + (2)4

-3 + 4 + 4 + 4

-3 + (3)4

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

tn = a + (n - 1) d

-3 1 5 9

-3

a a + d a + 2d a + 3d

-3 + 4

2.5.3

Arithmetic Sequences

tn = a + (n - 1) d

GeneralTerm

FirstTerm

Number or Position of the Term

Common Difference

2.5.4

The General Arithmetic Sequence

Given the sequence -5, -1, 3, …:a) Find the common difference.

d = t2 - t1

= (-1) - ( -5) = 4

Note: The common differencemay be found by subtractingany two consecutive terms.

b) Find t10 . tn = a + (n - 1) d

c) Find the general term . tn = a + (n - 1) d

d) Which term is equal to 63?

tn = a + (n - 1) d63 = - 5 + 4n - 472 = 4n18 = n t18 = 63

a = -5n = 10d = 4tn = ?

a = -5n = ?d = 4

a = -5n = ?d = 4tn = 63

t10 = -5 + (10 - 1) 4 = -5 + (9) 4t10 = 31

= -5 + (n - 1) 4 = -5 + 4n - 4 tn = 4n - 9

63 = -5 + (n - 1) 42.5.5

Finding the Terms of an Arithmetic Sequence

Find the number of terms in 7, 3, -1, - 5 …, -117 . tn = a + (n - 1) d

A pile of bricks is arranged in rows. The number of bricksin 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 = 7n = ?d = -4tn = - 117

a = 65n = ?d = - 6tn = 11

-117 = 7 + (n - 1) (-4) -117 = 7 - 4n + 4 -117 = -4n + 11 -128 = -4n 32 = n

11 = 65 + (n - 1) (-6)-60 = -6n 10 = n

a = 65n = ?d = - 6tn = 5

5 = 65 + (n - 1) (-6)-66 = -6n n = 11

2.5.6

Finding the Number of Terms of an Arithmetic Sequence

The 10th row contains 11 bricks.

There are 11 rows in this pile.

There are 32 terms inthe sequence.

Arithmetic means are 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

Therefore, the terms are:

a = 6n = 7d = ?tn = 30

30 = 6 + (7 - 1)d30 = 6 + 6d24 = 6d 4 = d

6, , 30 10, 14, 18, 22, 262.5.7

Arithmetic Means

Pages 74 - 761 - 43 odd46, 47, 49, 5052, 53, 56, 57 2.5.8

Suggested Questions: