6.01 Introduction to 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 $u_n$un​. For the sequence $2,3,5,7,11,...$2,3,5,7,11,... the sequence of prime numbers $u_1=2$u1​=2, $u_2=3$u2​=3, $u_3=5$u3​=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 die to generate a sequence 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 sequences

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 $u_n=\sqrt{n}$un​=√n means that the $n$nth term is the square root of $n$n. So the first term becomes $u_1=\sqrt{1}=1$u1​=√1=1  and the second term $u_2=\sqrt{2}$u2​=√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:

$u_n=u_{n-1}+4,u_1=5$un​=un−1​+4,u1​=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	$u_n$un​	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	$u_n$un​	$5+(n-1)\times4=1+4n$5+(n−1)×4=1+4n

 

Practice questions

Question 1

 

Question 2