6.2 Arithmetic Sequences
A sequence like 2, 5, 8, 11, … , where the difference
between consecutive terms is a constant, is called an
arithmetic sequence. In an arithmetic sequence, the
first term, t1, is denoted by the letter a. Each term
after the first is found by adding a constant, called
the common difference, d, to the preceding term.
I NVESTIGATE & I NQUIRE
For about 200 years, the Gatineau River was
used as a highway by logging companies. Logs
from the Canadian Shield were floated down
the river to the Ottawa River. It has been
estimated that 2% of the hundreds of millions
of logs that floated down the Gatineau sank.
Those that sank below the oxygen level are
perfectly preserved and are now being
harvested by water loggers, who wear scuba-diving gear.
The pressure that a diver experiences is the sum of the pressure of the
atmosphere and the pressure of the water. The increase in pressure with depth
under water follows an arithmetic sequence. If a diver enters the water when the
atmospheric pressure is 100 kPa (kilopascals), the pressure at a depth of 1 m is
about 110 kPa. At a depth of 2 m, the pressure is about 120 kPa, and so on.
1.
Copy and complete the table for this sequence.
t1
t2
Pressure (kPa)
100
110
Pressure (kPa) Expressed
Using 100 and 10
100
100 +1(10)
a
a+
Pressure Expressed
Using a and d
2.
Term
t3
120
t4
t5
130
140
What are the values of a and d for this sequence?
3. When you write an expression for a term using the letters a and d, you are
writing a formula for the term. What is the formula for t6? t8? t9?
436 MHR • Chapter 6
4.
Evaluate t8 and t9.
The Gatineau River has maximum depth of 35 m. What pressure
would a diver experience at this depth?
5.
EXAMPLE 1 Writing Terms of a Sequence
Given the formula for the nth term of an arithmetic sequence, tn = 2n + 1,
write the first 6 terms.
SOLUTION 1 Paper-and-Pencil Method
tn = 2n + 1
t1 = 2(1) + 1 = 3
t2 = 2(2) + 1 = 5
t3 = 2(3) + 1 = 7
t4 = 2(4) + 1 = 9
t5 = 2(5) + 1 = 11
t6 = 2(6) + 1 = 13
The first 6 terms are 3, 5, 7, 9, 11, and 13.
SOLUTION 2 Graphing-Calculator Method
Use the mode settings to choose the Seq
(sequence) graphing mode. Use the sequence
function from the LIST OPS menu to generate
the first 6 terms.
The first 6 terms are 3, 5, 7, 9, 11, and 13.
Note that the arithmetic sequence defined by tn = 2n + 1, or f(n) = 2n + 1,
in Example 1, is a linear function, as shown by the following graphs.
tn
12
10
8
6
4
2
0
2
4
6
n
6.2 Arithmetic Sequences • MHR 437
EXAMPLE 2 Determining the Value of a Term
Given the formula for the nth term, find t10.
a)
tn = 7 + 4n
b)
f(n) = 5n – 8
SOLUTION 1 Pencil-and-Paper Method
a)
tn = 7 + 4n
t10 = 7 + 4(10)
= 7 + 40
= 47
b)
f(n) = 5n – 8
f(10) = 5(10) – 8
= 50 – 8
= 42
SOLUTION 2 Graphing-Calculator Method
Use the mode settings to choose the Seq (sequence) graphing mode. Use the
sequence function from the LIST OPS menu to generate the 10th term.
a)
b)
Note that the general arithmetic sequence is
a, a + d, a + 2d, a + 3d, …
where a is the first term and d is the common difference.
t1 = a
t2 = a + d
t3 = a + 2d
..
.
tn = a + (n − 1)d, where n is a natural number.
Note that d is the difference between any successive pair of terms.
For example,
t2 − t1 = (a + d ) − a
=d
t3 − t2 = (a + 2d ) − (a + d )
= a + 2d − a − d
=d
438 MHR • Chapter 6
EXAMPLE 3 Finding the Formula for the nth Term
Find the formula for the nth term, tn , and find t19, for the arithmetic
sequence 8, 12, 16, …
SOLUTION
For the given sequence, a = 8 and d = 4.
Substitute known values:
Expand:
Simplify:
tn = a +(n − 1)d
= 8 + (n − 1)4
= 8 + 4n − 4
= 4n + 4
Three ways to find t19 are as follows.
Method 1
tn = a +(n − 1)d
t19 = a + (19 − 1)d
= a + 18d
= 8 + 18(4)
= 8 + 72
= 80
Method 2
tn = 4n + 4
t19 = 4(19) + 4
= 76 + 4
= 80
Method 3
Use a graphing calculator.
So, tn = 4n + 4 and t19 = 80.
EXAMPLE 4 Finding the Number of Terms
How many terms are there in the following sequence?
−3, 2, 7, … , 152
SOLUTION
For the given sequence, a = −3, d = 5, and tn = 152.
Substitute the known values in the formula for the general
term and solve for n.
6.2 Arithmetic Sequences • MHR 439
tn = a + (n − 1)d
152 = –3 + (n − 1)5
152 = –3 + 5n − 5
152 = 5n − 8
152 + 8 = 5n − 8 + 8
160 = 5n
160 5n
=
5
5
32 = n
Substitute known values:
Expand:
Simplify:
Solve for n:
The sequence has 32 terms.
EXAMPLE 5 Finding tn Given Two Terms
In an arithmetic sequence, t7 = 121 and t15 = 193. Find the first three
terms of the sequence and tn.
SOLUTION
Substitute known values in the formula for the nth term to write a system
of equations. Then, solve the system.
Write an equation for t7:
Write an equation for t15:
Subtract (1) from (2):
Solve for d:
Substitute 9 for d in (1):
Solve for a:
tn = a + (n − 1)d
121 = a + (7 − 1)d
121 = a + 6d
(1) The (1) shows that we are naming the
equation as “equation one.”
193 = a + (15 − 1)d
193 = a + 14d
(2)
72 = 8d
9=d
121 = a + 6(9)
121 = a + 54
67 = a
You can check by substituting 67 for a and 9 for d in (2).
Since a = 67 and d = 9, the first three terms of the sequence are 67, 76, and 85.
To find tn , substitute 67 for a and 9 for d in the formula for the nth term.
Simplify:
tn = a + (n − 1)d
tn = 67 + (n − 1)9
tn = 67 + 9n − 9
tn = 9n + 58
So, the first three terms are 67, 76, and 85, and tn = 9n + 58.
440 MHR • Chapter 6
Key
Concepts
• The general arithmetic sequence is a, a + d, a + 2d, a + 3d, … , where a
is the first term and d is the common difference.
• The formula for the nth term, tn or f(n), of an arithmetic sequence is
tn = a + (n – 1)d, where n is a natural number.
Communicate
Yo u r
Understanding
Given the formula for the nth term of an arithmetic sequence, tn = 4n – 3,
describe how you would find the first 6 terms.
2. a) Describe how you find the formula for the nth term of the arithmetic
sequence 3, 8, 13, 18, …
b) Describe how you would find t46 for this sequence.
3. Describe how you would find the number of terms in the sequence
5, 10, 15, … , 235.
4. Given that t5 = 11 and t12 = 25 for an arithmetic sequence, describe how
you would find tn for the sequence.
1.
Practise
A
1. Find the next three terms of each
arithmetic sequence.
a) 3, 7, 11, …
b) 33, 27, 21, …
c) −23, −18, −13, … d) 25, 18, 11, …
3 5 7
e) 5.8, 7.2, 8.6, …
f) , , , …
4 4 4
Given the formula for the nth term of an
arithmetic sequence, write the first 4 terms.
a) tn = 3n + 5
b) f(n) = 2n − 7
c) tn = 4n − 1
d) f(n) = 6 − n
n+3
e) tn = −5n − 2
f) f(n) = 2
2.
Given the formula for the nth term of an
arithmetic sequence, write the indicated term.
a) tn = 2n − 5; t11
b) tn = 4 + 3n; t15
c) f(n) = −4n + 5; t10 d) f(n) = 0.1n − 1; t25
3.
e)
2n − 1
tn = 2.5n + 3.5; t30 f) f(n) = ; t12
3
Determine which of the following
sequences are arithmetic. If a sequence is
arithmetic, write the values of a and d.
a) 5, 9, 13, 17, …
b) 1, 6, 10, 15, 19, …
c) 2, 4, 8, 16, 32, …
d) −1, −4, −7, −10, …
e) 1, −1, 1, −1, 1, …
1 2 3 4
f) , , , , …
2 3 4 5
g) −4, −2.5, −1, 0.5, …
2 3 4
h) y, y , y , y , …
i) x, 2x, 3x, 4x, …
j) c, c + 2d, c + 4d, c + 6d, …
4.
6.2 Arithmetic Sequences • MHR 441
Given the values of a and d, write the first
five terms of each arithmetic sequence.
a) a = 7, d = 2
b) a = 3, d = 4
c) a = −4, d = 6
d) a = 2, d = −3
1
5
e) a = −5, d = −8
f) a = , d = 2
2
g) a = 0, d = −0.25
h) a = 8, d = x
i) t1 = 6, d = y + 1
j) t1 = 3m, d = 1 − m
5.
Find the formula for the nth term and
find the indicated terms for each arithmetic
sequence.
a) 6, 8, 10, … ; t10 and t34
b) 12, 16, 20, … ; t18 and t41
c) 9, 16, 23, … ; t9 and t100
d) −10, −7, −4, … ; t11 and t22
e) −4, −9, −14, … ; t18 and t66
1 3 5
f) , , , … ; t12 and t21
2 2 2
g) 5, −1, −7, … ; t8 and t14
h) 7, 10, 13, … ; t15 and t30
i) 10, 8, 6, … ; t13 and t22
j) x, x + 4, x + 8, … ; t14 and t45
6.
Find the number of terms in each of the
following arithmetic sequences.
a) 10, 15, 20, … , 250
b) 1, 4, 7, … , 121
7.
40, 38, 36, … , −30
−11, −7, −3, … , 153
−2, −8, −14, … , −206
7
f) −6, − , −1, … , 104
2
g) x + 2, x + 9, x + 16, … , x + 303
c)
d)
e)
8. Find a and d, and then write the formula
for the nth term, tn , of arithmetic sequences
with the following terms.
a) t5 = 16 and t8 = 25
b) t12 = 52 and t22 = 102
c) t50 = 140 and t70 = 180
d) t2 = −12 and t5 = 9
e) t7 = −37 and t10 = −121
f) t8 = 166 and t12 = 130
g) t4 = 2.5 and t15 = 6.9
h) t3 = 4 and t21 = −5
The third term of an arithmetic sequence
is 24 and the ninth term is 54.
a) What is the first term?
b) What is the formula for the nth term?
9.
The fourth term of an arithmetic
sequence is 14 and the eleventh term is −35.
a) What are the first four terms?
b) What is the formula for the nth term?
10.
Apply, Solve, Communicate
11. The graph of an arithmetic sequence is shown.
a) What are the first five terms of the sequence?
b) What is t50? t200?
tn
70
60
50
40
30
20
10
0
442 MHR • Chapter 6
1 2 3 4 5
n
B
Find the common difference of the sequence whose formula for the nth
term is tn = 2n – 3.
12.
Copy and complete each arithmetic sequence. Graph tn versus n for each
sequence.
a) ■, ■, 14, ■, 26
b) ■, 3, ■, ■, −18
13.
14. Olympic Games The modern summer Olympic Games were first held
in Athens, Greece, in 1896. The games were to be held every four years, so
the years of the games form an arithmetic sequence.
a) What are the values of a and d for this sequence?
b) Research In what years were the games cancelled and why?
c) What are the term numbers for the years the games were cancelled?
d) What is the term number for the next summer games?
15. Multiples
How many multiples of 5 are there from 15 to 450, inclusive?
The 18th term of an arithmetic sequence is 262. The common difference
is 15. What is the first term of the sequence?
16.
Barrie is 60 km north of Toronto by road. If you drive north
from Barrie at 80 km/h, how far are you from Toronto by road after
a) 1 h?
b) 2 h?
c) t hours?
17. Driving
Comets approach the Earth at regular
intervals. For example, Halley’s Comet reaches its closest point to the Earth
about every 76 years. Comet Finlay is expected to reach its closest point to
the Earth in 2009, 2037, and three times between these years. In which years
between 2009 and 2037 will Comet Finlay reach its closest point to the
Earth?
18. Inquiry/Problem Solving
Amber works as an electrician. She charges $60 for each
service call, plus an hourly rate. If she charges $420 for an 8-h service call
a) what is her hourly rate?
b) how much would she charge for a 5-h service call?
19. Electrician
Franco is the manager of a health club. He earns a salary of
$25 000 a year, plus $200 for every membership he sells. What will he earn
in a year if he sells 71 memberships? 88 memberships?
104 memberships?
20. Salary
6.2 Arithmetic Sequences • MHR 443
Ring Size
A ring size indicates a standardized
1
inside diameter of a ring. The table gives the inside
2
diameters for 5 ring sizes.
3
a) Determine the formula for the nth term of
4
the sequence of inside diameters.
5
b) Use the formula to find the inside diameter of a size 13 ring.
21. Ring sizes
Inside Diameter (mm)
12.37
13.2
14.03
14.86
15.69
Boxes are stacked in a store display in the
shape of a triangle. The numbers of boxes in the rows form an arithmetic
sequence. There are 41 boxes in the 3rd row from the bottom. There are
23 boxes in the 12th row from the bottom.
a) How many boxes are there in the first (bottom) row?
b) What is the formula for the nth term of the sequence?
c) What is the maximum possible number of rows of boxes?
22. Displaying merchandise
On the first day of practice, the soccer team ran eight
40-m wind sprints. On each day after the first, the number of wind sprints
was increased by two from the day before.
a) What are the values of a and d for this sequence?
b) Write the formula for the nth term of the sequence.
c) How many wind sprints did the team run on the 15th day of practice?
How many metres was this?
23. Application
24. Pattern
How many dots are in the 51st figure?
•
•
•
•
•
n=1
•
•
2
•
•
•
•
•
•
•
•
3
•
4
25. Pattern The U-shapes are made from asterisks.
a) How many asterisks are in the 4th diagram?
b) What is the formula for the nth term of the sequence
in the numbers of asterisks?
c) How many asterisks are in the 25th diagram?
d) Which diagram contains 139 asterisks?
1
2
The time from one full moon to the next is 29.53 days.
The first full moon of a year occurred 12.31 days into the year.
a) How many days into the year did the 9th full moon occur?
b) At what time of day did the 9th full moon occur?
26. Astronomy
444 MHR • Chapter 6
3
The eighth term of an arithmetic sequence is 5.3 and the fourteenth
term is 8.3. What is the fifth term?
27.
Use finite differences to explain why the graph
of tn versus n for an arithmetic sequence is linear.
b) Explain why the points on a graph of tn versus n for an arithmetic
sequence are not joined by a straight line.
28. Communication a)
The period of a pendulum is the time it
takes to complete one back-and-forth swing. On the Earth, the period,
T seconds, is approximately given by the formula T = 2l, where
l metres is the length of the pendulum. If a 1-m pendulum completes
its first period at a time of 10:15:30, or 15 min 30 s after 10:00,
a) at what time would it complete 100 periods? 151 periods?
b) how many periods would it have completed by 10:30:00?
29. Motion of a pendulum
30. Motion of a pendulum
Repeat question 29 for a 9-m pendulum
on the Earth.
The period of a pendulum depends on the
acceleration due to gravity, so the period would be different on the moon
than on the Earth. On the moon, the period, T seconds, would be given
approximately by the formula T = 5l, where l metres is the length of the
pendulum. Repeat question 29 for a 1-m pendulum on the moon.
31. Motion of a pendulum
C
32. Measurement The side lengths in a right triangle form an arithmetic
sequence with a common difference of 2. What are the side lengths?
The sum of the first two terms of an arithmetic sequence is 16. The
sum of the second and third terms is 28. What are the first three terms of
the sequence?
33.
How does the sum of the first and fourth terms of an arithmetic
sequence compare with the sum of the second and third terms? Explain.
b) Find two other pairs of terms whose sums compare in the same way
as the two pairs of terms in part a).
34. a)
The first four terms of an arithmetic sequence are 4, 13, 22, and 31.
Which of the following is a term of the sequence?
316 317 318 319 320
35.
6.2 Arithmetic Sequences • MHR 445
The first term of an arithmetic sequence is represented by
3x + 2y. The third term is represented by 7x. Write the expression that
represents the second term.
36. Algebra
37. Algebra Determine the value of x that makes each sequence arithmetic.
a) 2, 8, 14, 4x, …
b) 1, 3, 5, 2x − 1, …
c) x − 2, x + 2, 5, 9, …
d) x − 4, 6, x, …
e) x + 8, 2x + 8, −x, …
Find the value of x so that the three given terms are
consecutive terms of an arithmetic sequence.
a) 2x – 1, 4x, and 5x + 3
b) x, 0.5x + 7, and 3x – 1
2
c) 2x, 3x + 1, and x + 2
38. Algebra
Find a, d, and tn for the arithmetic sequence with the terms
t7 = 3 + 5x and t11 = 3 + 23x.
39. Algebra
40. Algebra
Show that tn − tn − 1 = d for any arithmetic sequence.
A C H I E V E M E N T Check
Knowledge/Understanding Thinking/Inquiry/Problem Solving Communication Application
3, 14, 25, … and 2, 9, 16, … are two arithmetic sequences. Find the first ten
terms common to both sequences.
LOGIC
Power
A box contains 5 coloured cubes and an
empty space the size of a cube.
Use moves like those in checkers. In one move, one cube can slide to an empty
space or jump over one cube to an empty space. Find the smallest number of
moves needed to reverse the order of the cubes.
446 MHR • Chapter 6