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... and 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,... (the sequence of prime numbers) the first term is 2, the second term is 3 and term 3 is 5. We can show this with the following notation t_1=2, t_2=3, t_3=5, and so on. Term 6 of this sequence is 13, t_6=13.

To refer generally to the nth term we use the notation t_n. The n is a variable which represents the position of a term in the sequence. If we let n=1 then t_n is 1, if we let n=6 then t_n is t_6. For any term using this notation t_n, we can then see that the term before would have to be t_{n-1} and the term after t_n is t_{n+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,... by separating the digits of \pi, we could use a dice to generate a sequence of of rolls 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.

Examples

Example 1

If T_n describes the nth term in the following sequence, what is T_3?4,\,-5,\,6,\,-7,\,8,\,\ldots

Worked Solution

Create a strategy

Find the term in the third position from left to right.

Apply the idea

T_3 refers to the third term which is 6.T_3=6

Idea summary

A sequence is an ordered collection of numbers. 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. To refer generally to the nth term we use the notation t_n.

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

Note that t_n is the next term after t_{n-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 being 3". Therefore the second term t_2 is equal to twice the first term t_1 plus 2, which is 2\times 3+2 or 8.

The third term is: t_3=2\times t_2+3=19. The fourth term is: t_4=2\times t_3+4=42. This process of deducing the nth term from the \left(n-1\right)th term can continue indefinitely. And our sequence can be listed as 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.

Examples

Example 2

Which of the following are recurrence relations? Select all that apply.

\dfrac{S_n}{n}=n+1

R_n = \left( 3 R_{n - 1}\right)^{5} + R_{n - 2}

T_n = T_{n - 1} + 9

V_n = 3 \left(n - 1\right)

Worked Solution

Create a strategy

Choose the equations that show how the general nth value is related to the term before it, the (n-1)th term.

Apply the idea

Among the choices, only Option B: R_n = \left( 3 R_{n - 1}\right)^{5} + R_{n - 2} and Option C: T_n = T_{n - 1} + 9 are recurrence relations because their equations express the nth term in terms of the (n-1)th term.

Example 3

Using the following recursive rule, state the first 5 terms of the sequence in order.t_n = 2 t_{n - 1},\ t_1 = 2

Worked Solution

Create a strategy

Find t_1 to t_5 using the rule.

Apply the idea

We are given that t_1=2. We can use the equation t_n=2t_{n-1} to find the next 4 terms.

\displaystyle t_2	\displaystyle =	\displaystyle 2\times t_{1}	Substitute n=2
	\displaystyle =	\displaystyle 2\times 2	Substitute t_1=2
	\displaystyle =	\displaystyle 4	Evaluate
\displaystyle t_3	\displaystyle =	\displaystyle 2 \times t_{2}	Substitute n=3
	\displaystyle =	\displaystyle 2\times 4	Substitute t_2=4
	\displaystyle =	\displaystyle 8	Evaluate
\displaystyle t_4	\displaystyle =	\displaystyle 2 \times t_{3}	Substitute n=4
	\displaystyle =	\displaystyle 2\times 8	Substitute t_3=8
	\displaystyle =	\displaystyle 16	Evaluate
\displaystyle t_5	\displaystyle =	\displaystyle 2 \times t_{4}	Substitute n=5
	\displaystyle =	\displaystyle 2\times 16	Substitute t_4=16
	\displaystyle =	\displaystyle 32	Evaluate

So, the first 5 terms of the sequence are 2,\, 4,\, 8, \,16,\, 32.

Idea summary

We can express a sequence using a recurrence relation when each new term is generated by some function of a previous term or terms. The term t_n is usually expressed in terms of the previous term t_{n-1}.

A recursive rule always consists of two parts: the rule that shows how the sequence recurs and the initial term.

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... The recursive rule for this sequence is t_{n+1}=t_n+2,t_1=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}, 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. For this sequence the explicit rule would be t_n=2n+1. We can see if we wanted to find the 5th term we can substitute 5 into the place of n, t_5=2\times 5+1=11. By continuing the pattern seen in the sequence 3,5,7,9... we can see that the 5th term is 11. We can use this rule to jump to a later term like t_{43}:t_{43}=2\times 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.

Examples

Example 4

Using the following explicit rule, state the first 5 terms of the sequence in order starting with n=1.s_n=n^2+6

Worked Solution

Create a strategy

Substitute n=1,2,3,4,5 into the explicit rule.

Apply the idea

\displaystyle s_1	\displaystyle =	\displaystyle 1^2+6	Substitute n=1
	\displaystyle =	\displaystyle 7	Evaluate
\displaystyle s_2	\displaystyle =	\displaystyle 2^2+6	Substitute n=2
	\displaystyle =	\displaystyle 10	Evaluate
\displaystyle s_3	\displaystyle =	\displaystyle 3^2+6	Substitute n=3
	\displaystyle =	\displaystyle 15	Evaluate
\displaystyle s_4	\displaystyle =	\displaystyle 4^2+6	Substitute n=4
	\displaystyle =	\displaystyle 22	Evaluate
\displaystyle s_5	\displaystyle =	\displaystyle 5^2+6	Substitute n=5
	\displaystyle =	\displaystyle 31	Evaluate

So, the first 5 terms of the sequence are 7,\,10,\,15,\,22,\,31.

Idea summary

An explicit rule for t_n is in terms of n. Explicit rules allow us to find any term in any position without needing to know any of the terms before it.

Sequences with technology

A CAS calculator can be used to generate the terms of a sequence given a recursive or explicit rule. When problem solving involving sequences we can make effective use of our calculator to:

List the terms of the sequence
Find a particular term in a sequence
Calculate the sum of a given set of terms from a sequence
Graph the sequence to observe patterns in the behaviour of the sequence

Examples

Example 5

Consider the following sequence.T_n=T_{n-1}+5,T_1=10

Find T_{40}.

Worked Solution

Create a strategy

Use the sequence facility of your calculator or other technology.

Apply the idea

Enter the given sequence rule into your calculator, along with the initial term T_1 and find the 40th term where n=40.

T_{40}=205

Find the first term greater than or equal to 300.

Worked Solution

Create a strategy

Use the sequence facility of your calculator or other technology.

Apply the idea

Using your calculator, find the first term that is greater than or equal to 300 either by generating a list of terms or by graphing the sequence.

\text{Answer}=300

If S_n is the sum of the first n terms, find S_{30}.

Worked Solution

Create a strategy

Using calculator, sum the first 30 terms in the sequence.

Apply the idea

You can use your calculator to generate a list of the sum of n terms for up to n=30.

S_{30}=2475

Starting from n=1, find the minimum number of terms for the sum to be greater than 5000.

Worked Solution

Create a strategy

Using your calculator, trial and error the sum of different number of terms. Use the fact that it has to be at least 30 terms as S_{30}=2475 from part (c).

Apply the idea

\text{Minimum number of terms}=44

Example 6

Consider the following sequence starting at n=1.T_n=5-3n

Find T_{30}.

Worked Solution

Create a strategy

Enter the rule into your calculator, along with the initial term T_1 and scroll down to find the 30th term.

Apply the idea

T_{30}=-85

Find the first term less than -150.

Worked Solution

Create a strategy

Using your calculator, scroll down the sequence to find the first term that is less than -150.

Apply the idea

\text{Answer}=-151

If S_n is the sum of the first n terms, find S_{15}.

Worked Solution

Create a strategy

Using calculator, sum the first 15 terms in the sequence. There should be a sum function, where you can input n=15.

Apply the idea

S_{15}=-285

Starting from n=1, find the minimum number of terms required for the sum to be less than -600.

Worked Solution

Create a strategy

Using your calculator, trial and error the sum of different number of terms. Use the fact that it has to be at least 15 terms as S_{15}=-285 from part (c).

Apply the idea

\text{Minimum number of terms}=22

Idea summary

When problem solving involving sequences we can make effective use of our calculator to:

List the terms of the sequence
Find a particular term in a sequence
Calculate the sum of a given set of terms from a sequence
Graph the sequence to observe patterns in the behaviour of the sequence

3.01 Sequences

Ideas

Sequences

Examples

Example 1

Recurrence relations

Examples

Example 2

Example 3

Explicit rules

Examples

Example 4

Sequences with technology

Examples

Example 5

Example 6

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