5.01 Number patterns and sequences

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 simple 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 pattern, where 2 is added at every step.

This is a decreasing sequence, where 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.

Examples

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. The terms in a sequence are referred to using subscript numbers. For example the initial term is commonly labelled, t_0, and the first term, t_1, and the second term, t_2 and so on. In general, any nth number term in a sequence can be referred to as t_n and any previous term would be t_{n-1} and the subsequent term would be t_{n-1}.

Note that any letter could be used to refer to the terms, not just t.

If the sequence ends, it is known as a finite sequence. -3,5,13,21 and 1,10,100,1000,10000 are examples of finite sequences. An infinite sequence is a sequence with infinite terms, in other words a sequence that never ends. 1,1/2,1/3,1/4,1/5,1/6,... is an example of an infinite sequence, where one keeps being added to the denominator to create the next term.

The "dot dot dot" \ldots 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.

For example the rule t_n=\dfrac1{n} means that the nth term is the reciprocal of n. This is an example where n=1,2,3,..., so t_1 would be the first term and not t_0, as n=0 cannot be substituted into this generating rule. So the first term becomes t_1=\dfrac11=1 and the second term t_2=\dfrac12 etc., so that the sequence becomes 1,1/2,1/3,1/4,1/5,1/6,... and so on.

There are some sequences that appear to have no pattern at all and therefore is difficult or impossible to find an explicit generating rule. But they can nevertheless have a certain logical way of building. For example, the sequence 3,1,4,1,5,9,... separating the digits of \pi exhibits no discernible pattern and continues to do so indefinitely.

Examples

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:

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

Notice that the number of petals are increasing by 5 each time - in particular, the value of a_n is always equal to 5 times the value of n. Therefore, the generating rule for this sequence must be a_n=5n, where a_1=5 represents the first term.

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

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

Examples

Example 4

Matches were used to make the pattern attached:

a

Complete the table:

\text{Number of triangles } (t)	1	2	3	5	10	20
\text{Number of matches } (m)

Worked Solution

Create a strategy

Use the diagram to find the number of matches added every time the number of triangle increases.

Apply the idea

Using the diagram, we can count the number of matches that make up the 1,2, and 3 triangles.

\text{Number of triangles } (t)	1	2	3	5	10	20
\text{Number of matches } (m)	3	5	7

Since 3+2=5 and 5+2=7, this means that 2 matches are added every time to form a new triangle.

So if the diagram shows that 4 triangles have 9 matches, we have 9+2=11 matches that make up 5 triangles.

10 is double of 5 triangles so we need 11+10=21 matches, and 20 is double of 10 triangles so we need 21+20=41 matches.

\text{Number of triangles } (t)	1	2	3	5	10	20
\text{Number of matches } (m)	3	5	7	11	21	41

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b

Write a formula that describes the relationship between the number of matches, m, and the number of triangles, t.

Worked Solution

Create a strategy

Use the pattern found in part (a).

Apply the idea

We need to multiply the number of triangles by 2 which is the number of matches added every time and add 1 match to enclose the shape.

m= 2t+1

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c

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

Worked Solution

Create a strategy

Use the formula from part (b) using t=77.

Apply the idea

\displaystyle m	\displaystyle =	\displaystyle 2t+1	Write the formula
	\displaystyle =	\displaystyle 2\times 77+1	Substitute t=77
	\displaystyle =	\displaystyle 154+1	Evaluate the multiplication
	\displaystyle =	\displaystyle 155	Evaluate

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