17.08 Sequences

 

An ordered collection of numbers (or objects) is called a sequence or progression. It is similar to a set, but may contain repetition and the order of the numbers is important. For example $-3,5,13,21...$−3,5,13,21... and $1,10,100,1000...$1,10,100,1000... are two interesting mathematical progressions.

If the sequence ends, it is known as a finite sequence. If the sequence continues indefinitely it is said to be infinite.

The elements of a sequence are called terms. For the sequence $2,3,5,7,11,...$2,3,5,7,11,... (the sequence of prime numbers) the first term is $2$2, the second term is $3$3 and term $3$3 is $5$5. We can show this with the following notation $t_1=2$t1​=2, $t_2=3$t2​=3, $t_3=5$t3​=5, and so on. Term $6$6 of this sequence is $13$13, $t_6=13$t6​=13.

To refer generally to the $n$nth term we use the notation $t_n$tn​. The $n$n is a variable which represents the position of a term in the sequence. If we let $n=1$n=1 then $t_n$tn​ is $t_1$t1​, if we let $n=6$n=6 then $t_n$tn​ is $t_6$t6​. For any term using this notation $t_n$tn​, we can then see that the term before would have to be $t_{n-1}$tn−1​ and the term after $t_n$tn​ is $t_{n+1}$tn+1​.

Sequences do not necessarily have to have a pattern or simple rule for generating terms in the sequence. We could create the sequence $3,1,4,1,5,9,...$3,1,4,1,5,9,... by separating the digits of $\pi$π, we could use a dice to generate a sequence of of rolls $2,1,1,3,4,...$2,1,1,3,4,... or we could write down the prime numbers in order. However, many sequences have a generating rule that can be expressed as a formula, let's look at a couple of different ways of describing such sequences.

Explicit rules

Consider the sequence $3,5,7,9...$3,5,7,9... . It would be helpful to have a rule which allows us to find any term and does not rely on knowing the term before to find it.

This rule is called the explicit rule or the general rule and it is written in terms of $n$n. For this sequence the explicit rule would be $t_n=2n+1$tn​=2n+1. We can see if we wanted to find the $5$5th term we can substitute $5$5 into the place of $n$n, $t_5=2\times5+1=11$t5​=2×5+1=11. By continuing the pattern seen in the sequence $3,5,7,9$3,5,7,9 ... we can see that the $5$5th term is $11$11. We can use this rule to jump to a later term like $t_{43}$t43​: $t_{43}=2\times43+1=87$t43​=2×43+1=87.

 

Worked example

Write the explicit rule to describe the $n$nth term of the sequence $5,9,13,17,21,...$5,9,13,17,21,...

Do: 

To obtain the explicit rule it can sometimes be useful to look at the sequence as a table of values and try to see the connection between $n$n and the term:

$n$n	$t_n$tn​	Pattern
$1$1	$5$5	$5$5
$2$2	$9$9	$5+4$5+4
$3$3	$13$13	$5+2\times4$5+2×4
$4$4	$17$17	$5+3\times4$5+3×4
...
$n$n	$t_n$tn​	$5+(n-1)\times4=1+4n$5+(n−1)×4=1+4n

Hence, the sequence could be written as the explicit rule $t_n=1+4n$tn​=1+4n.

 

Practice questions

Question 1

Question 2