6. Number Patterns and Recursions

Lesson

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.

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$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

Find the next number in the sequence:

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

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, $t_0$`t`0, and the first term, $t_1$`t`1, and the second term, $t_2$`t`2 and so on. In general, any $n$`n`^{th} number term in a sequence can be referred to as $t_n$`t``n` and any previous term would be $t_{n-1}$`t``n`−1 and the subsequent term would be $t_{n+1}$`t``n`+1

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

If the sequence ends, it is known as a finite sequence. $-3,5,13,21$−3,5,13,21 and $1,10,100,1000,10000$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,...$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" $...$... 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 $t_n=\frac{1}{n}$`t``n`=1`n` means that the $n$`n`th term is the reciprocal of $n$`n`. This is an example where $n=1,2,3,...$`n`=1,2,3,... , so $t_1$`t`1 would be the first term and not $t_0$`t`0 , as $n=0$`n`=0 cannot be substituted into this generating rule. So the first term becomes $t_1=\frac{1}{1}=1$`t`1=11=1 and the second term $t_2=\frac{1}{2}$`t`2=12 etc., so that the sequence becomes $1,1/2,1/3,1/4,1/5,1/6,...$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,...$3,1,4,1,5,9,... separating the digits of $\pi$π exhibits no discernible pattern and continues to do so indefinitely.

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

Finite

AInfinite

BFinite

AInfinite

B

State the first five terms of the sequence $a_n=3n-3$`a``n`=3`n`−3.

Write all five terms on the same line separated by a comma.

State the first five terms of the sequence $a_n=2^{n+1}$`a``n`=2`n`+1.

Write all five terms on the same line separated by a comma.

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 $a_n$an |
$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 $a_n$`a``n` 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 $a_n$`a``n` is always equal to $5$5 times the value of $n$`n`. Therefore, the generating rule for this sequence must be $a_n=5n$`a``n`=5`n`, where $a_1=5$`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$10 flowers present, substitute $n=10$`n`=10 into the rule to find $a_{10}=5\times10=50$`a`10=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 $a_n$`a``n` versus the number of flowers $n$`n` is shown in the graph below.

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`×2−4B$A=N^2-4$

`A`=`N`2−4C$A=2+N-4$

`A`=2+`N`−4D$A=2\times\left(N-4\right)$

`A`=2×(`N`−4)A$A=N\times2-4$

`A`=`N`×2−4B$A=N^2-4$

`A`=`N`2−4C$A=2+N-4$

`A`=2+`N`−4D

Matches were used to make the pattern attached:

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?

Define and explain key concepts in the concept of a sequence as a function, and apply a range of related mathematical routines and procedures

Define and explain key concepts in tabular and graphical display of sequences, and apply a range of related mathematical routines and procedures