8.01 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 order 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. Can you notice a pattern? What is similar and what is different about the patterns? If the sequence ends, it is known as a finite sequence. Otherwise it is said to be infinite.

The elements of a sequence are called terms and the $n$nth term is denoted by $t_n$tn​. For the sequence $2,3,5,7,11,...$2,3,5,7,11,... the sequence of prime numbers $t_1=2$t1​=2, $t_2=3$t2​=3, $t_3=5$t3​=5, and so on.

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.

Recurrence relations

We can express a sequence using a recurrence relation when each new term is generated by some function of a previous term or terms. Take for example, the sequence described by:

$a_n=2a_{n-1}+n,a_1=3$an​=2an−1​+n,a1​=3

This expression tells us that the first term $a_1$a1​ is $3$3. Then it shows us how to find, progressively, any other term in the sequence. For example, looking at the formula, the second term $a_2$a2​ is equal to twice the first term $a_1$a1​ plus $2$2, which is $2\times3+2$2×3+2 or $8$8.

The third term is: $a_3=2\times a_2+3=19$a3​=2×a2​+3=19.

The fourth term is:  $a_4=2\times a_3+4=42$a4​=2×a3​+4=42. This process of deducing the $n$nth term from the $\left(n-1\right)$(n−1)th term can continue indefinitely. And our sequence can be listed as $3,8,19,42,...$3,8,19,42,...

Explicit sequence

The terms in some sequences can be written as an explicit rule or formula in terms of $n$n, these generating rules allow the calculation of any particular term in the sequence. For example the rule $a_n=\sqrt{n}$an​=√n means that the $n$nth term is the square root of $n$n. So the first term becomes $a_1=\sqrt{1}=1$a1​=√1=1  and the second term $a_2=\sqrt{2}$a2​=√2  etc., so that the sequence becomes   $1,\sqrt{2},\sqrt{3},2,...$1,√2,√3,2,...and so on. 

Worked example

Find a recurrence relation and explicit rule to describe the $n$nth term of the sequence $5,9,13,17,21,...$5,9,13,17,21,... 

Think: For a recurrence relation we need to state the initial term and how to get the from one term to the next. 

Do: The initial term here is $5$5 and each term differs from the last by adding $4$4. So the recurrence relation is:

$t_n=t_{n-1}+4,t_1=5$tn​=tn−1​+4,t1​=5

To obtain the explicit rule it can sometimes be useful to look of 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 recurrence relation $t_n=t_{n-1}+4,t_1=5$tn​=tn−1​+4,t1​=5 or as the explicit rule $t_n=1+4n$tn​=1+4n.

Practice questions

Question 1

Question 2

Question 3

Question 4

Question 5

 

Fibonacci sequence

The Fibonacci sequence is where, apart from the first two terms, each term is the sum of the previous two terms. That is: $F_n=F_{n-1}+F_{n-2}$Fn​=Fn−1​+Fn−2​. By definition $F_0=0$F0​=0 and $F_1=1$F1​=1 but in many descriptions, the $0$0th term is left out and the starting definition is $F_1=F_2=1$F1​=F2​=1. This produces the following sequence:

$1,1,2,3,5,8,13,21,34,55,89,...$1,1,2,3,5,8,13,21,34,55,89,...

It also possesses some unusual mathematical properties such as:

• The sum of the squares of any two consecutive terms is another Fibonacci term.
• The ratio of consecutive terms, namely $\frac{3}{2},\frac{5}{3},\frac{8}{5},\frac{13}{8}...$32​,53​,85​,138​... approaches the golden ratio $\phi$ϕ where $\phi=\frac{1+\sqrt{5}}{2}=1.618033...$ϕ=1+√52​=1.618033...

• Apart from $F_1$F1​, $F_2$F2​ and $F_4$F4​ every prime term is sitting in a prime position. That is, if the $n$nth term $F_n$Fn​ is prime then $n$n must be prime.
• There are a surprising number of applications and occurrences in nature of the Fibonacci numbers, including computer search algorithms, tree branching, population growth as well as spirals on pine cones, pineapples, sea shells and ferns. Can you find some unexpected appearances?

 

The rule for the $n$nth term is more complex to derive, but in case you're curious, it is given by:

$F_n=\frac{1}{\sqrt{5}}\left(\phi^n-\left(-\phi\right)^{-n}\right),\ \phi=\frac{1+\sqrt{5}}{2}$Fn​=1√5​(ϕn−(−ϕ)−n), ϕ=1+√52​

 

Try creating a Fibonacci spiral of your own as follows:

Start with a blank piece of graph paper, and highlight a square with side length 1, somewhere towards the middle.

 

 

 

 

 

 

 

Next to that, highlight another square of side length 1.

 

 

 

 

 

 

 

 

Now, highlight a square of side length 2. Use the two squares from before as one of the sides. (like this image).

 

 

 

 

 

 

 

 

Next, build a square of length $3$3 on the side of the squares we have already built, then a side of length $5$5 and another of length $8$8. It should start to look like what I have here.

The final step is draw in a the spiral. We construct the spiral but joining diagonally opposite corners in a swooping arc motion. Start in the centre with the first square and then join corner to corner. This may take a little bit of practice to get the arcs looking nice.

And so you have constructed a Fibonacci spiral.

 

 

 

 

Practice questions

Question 6

Question 7

Question 8