You will need
4.3 The General Term of
• counters
a Sequence
GOAL
Write an algebraic expression for the general term of a sequence.
Learn about the Math
Maria is using counters to make a sequence
of rectangular figures. The first three figures
have 6, 9, and 12 counters.
many counters would
? How
Maria need to make the
100th figure?
Example 1: Using systematic trial
Determine a pattern rule for the sequence 6, 9, 12, 15, ….
Maria’s Solution
Term
number
Term
value
1
6
2
9
3
12
4
15
The pattern rule is 3t 3,
where t is the term number.
130 Chapter 4
I made a table of values using the first few terms in
the sequence.
Then I used guess and test to find the pattern rule with
variable t.
First I tried 6t because when t 1, 6t is 6(1) 6.
However, 6t doesn’t work when t 2 because 6(2) 9.
Next I tried 5t 1 because when t 1, you get
5(1) 1 6. But 5t 1 doesn’t work when t 2 because
5(2) 1 9.
Next I tried 4t 2. It works when t 1, but not when
t 2 because 4(2) 2 9.
Then I tried 3t 3. It works when t 1. It also works
when t 2 because 3(2) 3 9, when t 3 because
3(3) 3 12, and when t 4 because 3(4) 3 15.
NEL
Example 2: Using logical reasoning
Determine a pattern rule for the sequence 6, 9, 12, 15, ….
Toma’s Solution
Term
number
Term
value
1
6
2
9
3
12
4
15
3
3
3
If t represents the term
number, the rule is 3t 3.
I used a table of values. I noticed that when you go
down the second column, you add 3 more counters
for each new row.
I know that repeated addition is the same as multiplication.
So, I know that the pattern rule has to include 3t.
If I try 3t by itself as the pattern rule, I get 3, 6, 9, 12, ….
This is not corrrect because each term is too low by 3.
If I add 3, I’ll have the right pattern rule: 3t 3.
I checked my pattern rule on all the term numbers,
and it works.
Using the pattern rule 3t 3, the value of the 100th term is
Communication Tip
3(100) 3 300 3
303
A pattern rule that
uses a variable, such
as n, to describe term
values in a sequence
is usually called the
nth term of the
sequence. It is also
called the general
term.
Maria would need 303 counters to make the 100th figure.
Reflecting
1. Why is it useful to create an algebraic expression for the general
term of a sequence?
2. What strategies can you use to create an algebraic expression for the
general term of a sequence?
Work with the Math
Example 3: Representing the n th term of a sequence
Determine an algebraic expression for the nth term of the sequence 2, 6, 10, 14, ….
Solution
NEL
Term number
Term value
1
2
2
6
3
10
4
14
4
4
4
The term value increases by 4 each time.
If you use the expression 4n, you get the sequence
4, 8, 12, 16, …. Each term is too high by 2.
So, the correct expression for the nth term is 4n 2.
When you check this expression using each term
number, you get the original sequence: 2, 6, 10, 14, ….
Patterns and Relationships
131
A
6. a) Write an algebraic expression for the
nth term of this sequence.
Checking
3. a) Copy and complete the table of values
for the sequence shown.
Term number
(figure
number)
Picture
1
Term value
(number
of squares)
2
2
3
4
5
b) Create an algebraic expression for the
nth term of the sequence.
c) Use your algebraic expression to
calculate the 30th term in the sequence.
B
Practising
b) Write an algebraic expression for the
general term of the sequence.
c) Use your expression for the general
term to calculate the 25th term in the
sequence.
5. Hendryk and Nilay are looking at this table
of values. Hendryk says that the pattern
rule is 2n 3. Nilay says that the pattern
rule is 3n 2. Which student is right?
Explain your thinking.
Term number
Term value
1
5
2
7
3
9
4
11
5
13
Figure 2
Figure 3
b) Calculate the number of counters you
would need to make the 80th figure in
this sequence.
c) Determine the figure number of the Z
you could make using 41 counters.
7. Vanya says that the 15th figure in this
sequence contains 225 small triangles.
Is she correct? Explain your thinking.
Figure 1
4. a) Make a table of values for the sequence
6, 11, 16, 21, 26, ….
132 Chapter 4
Figure 1
Figure 2
Figure 3
8. Benjamin made a rectangular sequence
using coloured counters.
Figure 1
Figure 2
Figure 3
a) Describe the pattern rule in words.
b) Write an algebraic expression for the
general term of the sequence.
c) Determine the figure number of the
rectangle you could make using 50
counters.
d) Calculate the number of blue
counters you would need to make the
75th rectangle in the sequence.
NEL
9. Write an algebraic expression for the nth
term of each sequence. Then use your
expression to calculate the 50th term in
the sequence.
a)
b)
c)
d)
e)
5, 9, 13, 17, 21, …
11, 13, 15, 17, 19, …
26, 31, 36, 41, 46, …
10, 40, 70, 100, 130, …
101, 201, 301, 401, 501, …
10. a) Start with the 5th row of a multiplication
table (the multiplication-by-5 row). Add
3 to each number in the row.
b) Determine an algebraic pattern rule for
the sequence in part (a).
c) Choose a different row of the
multiplication table. Add (or subtract) a
number of your choice to each number
in the row. Then determine an algebraic
pattern rule for the resulting sequence.
d) Discuss how the sequences in parts
(b) and (c) are similar to those in
question 9.
11. a) The terms of a sequence increase by
the same amount. The 1st term is 7,
and the 3rd term is 15. Calculate the
7th term. Then write an algebraic
expression for the general term.
b) The terms of a different sequence also
increase by the same amount. The
3rd term is 7, and the 7th term is 15.
Calculate the 17th term. Then write an
algebraic expression for the general term.
C
Extending
12. Write an algebraic expression for the nth
term of each sequence. Then use your
expression to calculate the 50th term in
the sequence.
13. a) Write an algebraic expression for the
nth term of this sequence.
b) Use your expression to predict the
number of blue counters in the 100th
figure.
c) Describe another strategy you could
use to predict the number of blue
counters in the 100th figure.
14. a) Make a table of values that shows the
total number of cubes in the first five
figures of this staircase pattern.
b) Write an algebraic pattern rule.
c) Calculate the number of cubes you
would need to build the 10th figure in
this staircase pattern.
15. Consider the staircase pattern in question 14.
a) Imagine a staircase with many rows
in this pattern. Use your imaginary
staircase to make a new pattern, in
which the 1st term is the number of
cubes in the 1st row, the 2nd term is
the number of cubes in the 2nd row,
and so on. Write the first six terms in
your new pattern.
b) Suppose that you made the original
staircase pattern out of cubes. Could
you split this pattern to show an adding
strategy to calculate the total number of
cubes? If so, explain how.
a) 29, 27, 25, 23, 21, …
b) 118, 117, 116, 115, 114, …
NEL
Patterns and Relationships
133