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.
We refer to the nth term as $t_n$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. If we think of 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.
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:
$t_n=2t_{n-1}+n,t_1=3$tn=2tn−1+n,t1=3
Note that $t_n$tn is the next term after $t_{n-1}$tn−1. We can describe the rule in words as "the next term is two times the previous term plus the term number ; with the first term $t_1$t1 being $3$3". Therefore the second term $t_2$t2 is equal to twice the first term $t_1$t1 plus $2$2, which is $2\times3+2$2×3+2 or $8$8.
The third term is: $t_3=2\times t_2+3=19$t3=2×t2+3=19.
The fourth term is: $t_4=2\times t_3+4=42$t4=2×t3+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,...
A recursive rule always consists of two parts. Firstly how the sequence recurs (how the next term is made) and secondly a term in which to start with, usually term $1$1.
Explicit rules
The recursive rule is limited in that it relies on the previous term in order to find the next term. Consider the sequence $3,5,7,9...$3,5,7,9... The recursive rule for this sequence is $t_{n+1}=t_n+2,t_1=3$tn+1=tn+2,t1=3. We can continue the sequence and find later terms by following this recursive rule however if we wanted to know a much later term, $t_{43}$t43, for example, 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 $t_n=3+2(n-1)$tn=3+2(n−1). We can see if we wanted to find the $5$5th term we can substitute $5$5 into the place of $n$n, $t_5=3+2(5-1)=11$t5=3+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; $t_{43}=3+2(43-1)=87$t43=3+2(43−1)=87.
If a question asks for a rule or equation, it will mean the explicit rule, if a question asks for a recurrence equation/rule then we use the recurrence relation.
Worked example
Write 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 from one term to the next term. We can describe the pattern in words as "next term is previous term plus $4$4; with term $1$1 equal to $5$5".
Do: We can use $t_n$tn for the next term after $t_{n-1}$tn−1. 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 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 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
If $T_n$Tn describes the $n$nth term in the following sequence, what is $T_3$T3?
$4,-5,6,-7,8,\ldots$4,−5,6,−7,8,…
Question 2
Which of the following are recurrence relations? Select all that apply.
$\frac{S_n}{n}=n+1$Snn=n+1
A$R_n=\left(3R_{n-1}\right)^5+R_{n-2}$Rn=(3Rn−1)5+Rn−2
B$T_n=T_{n-1}+9$Tn=Tn−1+9
C$V_n=3\left(n-1\right)$Vn=3(n−1)
D
Question 3
Which of the following are explicit relations? Select all that apply.
$b_n=b_{n-1}b_{n-2}$bn=bn−1bn−2
A$d_n=n^2+3n+8$dn=n2+3n+8
B$a_n=8a_{n-1}+a_1$an=8an−1+a1
C$c_n=3n^2$cn=3n2
D
Question 4
Using the following recursive rule, state the first $5$5 terms of the sequence in order.
$t_n=2t_{n-1}$tn=2tn−1, $t_1=2$t1=2
Enter each term on the same line, separated by commas.
Question 5
Using the following explicit rule, state the first $5$5 terms of the sequence in order starting with $n=1$n=1.
$s_n=n^2+6$sn=n2+6
Enter each term on the same line, separated by commas.