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Grade 9

6.01 Linear number sequences

Lesson

What is a sequence?

A sequence is a series of numbers that have an order and have a specific number pattern in the order. Getting from one number to the next can be thought of as a step. A linear pattern is formed when the same number is added or subtracted at each step. Let's take a look at the examples below:

This is an increasing (or growing)  sequence, where $2$2 is added at every step

This is a decreasing (or shrinking) sequence, where $3$3 is subtracted at every step

To find the next number that follows in a sequence, it's as simple as finding the pattern and applying to the last number. For example, the next number in the decreasing sequence above would be $5-3=2$53=2.

Practice questions

Question 1

Which of these options show an increasing pattern?

  1. $7,6,11,3,9$7,6,11,3,9

    A

    $6.5,6.6,6.7,6.8,6.9$6.5,6.6,6.7,6.8,6.9

    B

    $25,22,19,16,13$25,22,19,16,13

    C

QUESTION 2

Find the next number in the sequence:

  1. $7$7, $9$9, $11$11, $13$13, $\editable{}$

Describing sequences

In mathematics, a sequence is often given as a list of numbers, each separated by commas. Each of the separate numbers in a sequence can be called a term.

If the sequence ends, it is known as a finite sequence. $-3,5,13,21$3,5,13,21 and $1,10,19,28,37$1,10,19,28,37 are examples of finite sequences. An infinite sequence is a sequence with infinite terms, in other words a sequence that never ends. $4,9,14,19,24,...$4,9,14,19,24,... is an example of an infinite sequence, where five keeps being added to create the next term. The "dot dot dot" $...$... at the end of the sequence means that the number pattern in the sequence continues indefinitely, making this an infinite sequence versus a finite sequence.

 

When a number pattern is detectable in a progression, a generating rule can often be established and then used to determine a term in the sequence. Mathematicians can sometimes develop explicit generating rules that allow the calculation of any particular term in the sequence.

Practice questions

QUESTION 3

Is the sequence $1,2,3,4,5,6$1,2,3,4,5,6 finite or infinite?

  1. Finite

    A

    Infinite

    B

 
 

Using tables and graphs to describe a pattern

Below is a drawing of a simple pattern:

A table of values can be generated to count the number of petals visible at a given time, based on how many flowers are present:
 
Flowers    $n$n $1$1 $2$2 $3$3 $4$4
Petals     $p$p $5$5 $10$10 $15$15 $20$20

 

In this pattern, $n$n represents the step number in the sequence (that is, the number of flowers) and $p$p represents the total number of petals at that step.

Notice that the number of petals are increasing by $5$5 each time - in particular, the value of $p$p is always equal to $5$5 times the value of $n$n. Therefore, the generating rule for this sequence must be $p=5n$p=5n, where $p=5$p=5 represents the first term. This is a linear rule because the value of $p$p increases by the same amount each time. 

This rule can now be used to predict future results. For example, to calculate the total number of petals when there are $10$10 flowers present, substitute $n=10$n=10 into the rule to find $p=5\times10=50$p=5×10=50 petals. So even though there were only $1,2,3$1,2,3 and $4$4 flowers present in the sequence above, the rule has determined that there would be $50$50 petals visible when there are $10$10 flowers present.

A graph can also be used to represent a sequence. Consider the number pattern above, the number of petals $p$p versus the number of flowers $n$n is shown in the graph below.

Since the original pattern was a growing, linear pattern it is no surprise that the dots fall in an increasing straight line.

 

Practice questions

Question 4

Use the rule to complete the table of values:

"The starting number is doubled, then $4$4 is subtracted."

  1. Starting Number ($N$N) $12$12 $13$13 $14$14 $15$15
    Answer ($A$A) $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
  2. Which of these equations describes the rule above?

    $A=2\times\left(N-4\right)$A=2×(N4)

    A

    $A=N\times2-4$A=N×24

    B

    $A=N^2-4$A=N24

    C

    $A=2+N-4$A=2+N4

    D

Question 5

Matches were used to make the pattern attached:

  1. Complete the table:

    Number of triangles ($t$t) $1$1 $2$2 $3$3 $5$5 $10$10 $20$20
    Number of matches ($m$m) $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
  2. Write a formula that describes the relationship between the number of matches, $m$m, and the number of triangles, $t$t.

  3. How many matches are required to make $77$77 triangles using this pattern?

Outcomes

9.C1.2

Create algebraic expressions to generalize relationships expressed in words, numbers, and visual representations, in various contexts.

9.C3.2

Represent linear relations using concrete materials, tables of values, graphs, and equations, and make connections between the various representations to demonstrate an understanding of rates of change and initial values.

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