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$5−3=2.
Practice questions
Question 1
Which of these options show an increasing pattern?
$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:
$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?
Finite
AInfinite
B
Using tables and graphs to describe a pattern
Below is a drawing of a simple pattern:
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| 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.
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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$4 is subtracted."
Starting Number ($N$N) $12$12 $13$13 $14$14 $15$15 Answer ($A$A) $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Which of these equations describes the rule above?
$A=2\times\left(N-4\right)$A=2×(N−4)
A$A=N\times2-4$A=N×2−4
B$A=N^2-4$A=N2−4
C$A=2+N-4$A=2+N−4
D
Question 5
Matches were used to make the pattern attached:
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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) $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Write a formula that describes the relationship between the number of matches, $m$m, and the number of triangles, $t$t.
How many matches are required to make $77$77 triangles using this pattern?