Sequences and Recurrence Relations Revision Notes for VCE SSCE General Mathematics

Sequences and Recurrence Relations

What is a sequence?

A sequence is an ordered list of numbers written one after another. Each individual number in the sequence is called a term. We separate terms using commas when writing them out.

When a sequence continues indefinitely, or when there are too many terms to write completely, we use an ellipsis (three dots: ...) to show the pattern continues.

For example: $1, 3, 5, 7, 9, ...$

infoNote

The ellipsis notation (...) is a mathematical convention that indicates the pattern continues in the same way. It saves space and helps focus on understanding the pattern rather than writing out every term.

Generating sequences recursively

Sequences can be created randomly, but more commonly they follow a pattern using recursion. Recursion means we repeatedly apply the same rule to generate each new term from the previous one.

Understanding the recursive process

To create a sequence recursively, we need two things:

A starting value (the first term)
A rule that tells us how to get from one term to the next

chatImportant

The Two Essential Parts of Recursion

Every recursive sequence requires BOTH components:

A starting value to begin the sequence
A rule to generate each subsequent term

Missing either component means you cannot generate the sequence!

Example: The sequence $1, 3, 5, 7, 9, ...$ has:

Starting value: $1$
Rule: add $2$ to each term

We can visualize this as:

$1 \xrightarrow{+2} 3 \xrightarrow{+2} 5 \xrightarrow{+2} 7 \xrightarrow{+2} 9 \xrightarrow{+2} ...$

Worked example: Simple recursive rule

lightbulbExample

Worked Example: Creating a Sequence with a Simple Rule

Let's create the first five terms of a sequence with starting value $6$ and the rule "add $4$ to the previous term".

Step 1: Record the starting value: $6$

Step 2: Apply the rule (add $4$ ):

$6 + 4 = 10$

Step 3: Continue applying the rule three more times:

$10 + 4 = 14$

$14 + 4 = 18$

$18 + 4 = 22$

Answer: The first five terms are $6, 10, 14, 18, 22$

Worked example: Complex recursive rule

lightbulbExample

Worked Example: Applying a Multi-Step Rule

Now let's try a sequence with a more complex rule. Starting value is $5$ and the rule is "double the number and then subtract $3$ ".

Step 1: Record the starting value: $5$

Step 2: Apply the rule (multiply by $2$ , then subtract $3$ ):

$5 \times 2 - 3 = 7$

Step 3: Continue applying the same rule:

$7 \times 2 - 3 = 11$

$11 \times 2 - 3 = 19$

$19 \times 2 - 3 = 35$

Answer: The first five terms are $5, 7, 11, 19, 35$

infoNote

Notice how with complex rules, each step involves multiple operations. The key is to apply the ENTIRE rule to each term before moving to the next one. Don't rush - complete all operations in the rule before recording the new term.

Using a calculator to generate sequences

Generating sequences involves repetitive calculations - the same operation performed over and over. Calculators are excellent for this because they automatically store the previous answer and can reapply the same rule repeatedly.

Calculator method

When you perform a calculation on a calculator, it stores the result. You can then apply a new operation to that stored answer without retyping it.

lightbulbExample

Worked Example: Using a Calculator to Generate Sequences

Generate the first five terms with starting value $5$ and rule "double and subtract $3$ "

Step 1: Enter the starting value $5$ and press enter (or EXE)

Step 2: Enter the rule $\times 2 - 3$ and press enter

The calculator shows the result $7$ and remembers this value

Step 3: Press enter repeatedly

Each press applies the rule $\times 2 - 3$ to the previous result

Result: The sequence $5, 7, 11, 19, 35$ appears on screen

On a TI-Nspire calculator, you'll see calculations like " $5 \cdot 2-3$ " displayed. On a ClassPad, it shows "ans $\times 2-3$ " where "ans" represents the answer from the previous calculation.

infoNote

Different calculator models may display the calculations slightly differently, but the underlying principle is the same: the calculator stores each result and applies the rule to that stored value when you press enter again.

Recurrence relations

A recurrence relation is a formal mathematical way to describe how to generate a sequence. It consists of two essential parts:

A starting value - the first term in the sequence
A rule - describing how to calculate the next term from the current term

Writing recurrence relations in words

Consider the sequence: $10, 15, 20, ...$

In words, the recurrence relation is:

Start with $10$
To get the next term, add $5$ to the current term

Converting to symbolic notation

Rather than writing everything in words, mathematicians use a more compact symbolic form. We define a subscripted variable to represent the terms. The letter $V$ is commonly used, but any letter works.

Let $V_n$ be the term in the sequence after $n$ iterations (applications of the rule).

Here's how the word description translates to symbols:

Using this notation, we write the complete recurrence relation as:

$V_0 = 10, \quad V_{n+1} = V_n + 5$

This reads as:

$V_0 = 10$ : the starting value (before any iterations) is $10$
$V_{n+1} = V_n + 5$ : the next term equals the current term plus $5$

chatImportant

Understanding $V_0$ Notation

The starting value uses subscript $0$ (written as $V_0$ ) because at the start, we haven't yet applied the rule - there have been zero iterations.

$V_0$ = starting value (0 iterations)
$V_1$ = after applying the rule once (1 iteration)
$V_2$ = after applying the rule twice (2 iterations)
And so on...

This is why we start counting from 0, not 1!

Worked example: Generating from a recurrence relation

lightbulbExample

Worked Example: Finding Terms from a Recurrence Relation

Find the first five terms of the sequence defined by:

$V_0 = 29, \quad V_{n+1} = V_n - 4$

Step 1: Record the starting value:

$V_0 = 29$

Step 2: Use the rule to find $V_1$ :

$V_1 = V_0 - 4 = 29 - 4 = 25$

Step 3: Apply the rule to find three more terms:

$V_2 = V_1 - 4 = 25 - 4 = 21$

$V_3 = V_2 - 4 = 21 - 4 = 17$

$V_4 = V_3 - 4 = 17 - 4 = 13$

Answer: The first five terms are $29, 25, 21, 17, 13$

Using a calculator for recurrence relations

Calculators can generate sequences from recurrence relations very efficiently.

lightbulbExample

Worked Example: Calculator Method for Recurrence Relations

Generate the sequence from $V_0 = 300, \quad V_{n+1} = 0.5V_n - 9$ and determine how many terms are positive.

Steps:

Start with a blank screen
Enter $300$ and press enter
Enter $\times 0.5 - 9$ and press enter to generate the next term
Continue pressing enter until a negative term appears
Count the positive terms

Calculator output:

Table

Answer: The first five terms of the sequence are positive.

Naming and numbering terms in a sequence

When working with recurrence relations, each term has a specific name using subscript notation.

lightbulbExample

Worked Example: Identifying Terms by Subscript

Consider $V_0 = 3, \quad V_{n+1} = V_n + 6$

The sequence generated is: $3, 9, 15, 21, 27, 33, ...$

We label each term:

Value	$3$	$9$	$15$	$21$	$27$	$33$
Name	$V_0$	$V_1$	$V_2$	$V_3$	$V_4$	$V_5$

So if asked to find:

$V_1 = 9$
$V_4 = 27$
$V_5 = 33$

The subscript tells us which position the term occupies in the sequence, counting from $V_0$ as the starting position.

bookmarkSummary

Key Points to Remember:

A sequence is an ordered list of numbers; each number is called a term
Recursion means repeatedly applying the same rule to generate the next term from the previous one
To create a sequence recursively, you need a starting value and a rule
Calculators can generate sequences efficiently by storing the previous answer and reapplying the rule
A recurrence relation has the form $V_0 = \text{(starting value)}, \quad V_{n+1} = \text{(rule involving } V_n)$
Terms are numbered using subscripts: $V_0$ is the starting value, $V_1$ is after one iteration, $V_2$ after two iterations, and so on

Sequences and Recurrence Relations (VCE SSCE General Mathematics): Revision Notes