Number Patterns (VCE SSCE General Mathematics): Revision Notes

Number Patterns

What are sequences?

A sequence is an ordered arrangement of numbers or symbols that follows a specific pattern. The individual numbers or items within a sequence are called terms. When writing sequences, each term is separated by a comma.

When a sequence continues indefinitely, or when there are too many terms to write them all out, we use an ellipsis (three dots: ...) after the first few terms to indicate the pattern continues. For example:

$7, 3, 4, 11, 15, 24, \ldots$

This notation tells us the sequence keeps going according to the same pattern.

Types of sequences

Sequences can be created in two main ways: randomly or through recursion.

Random sequences

Random sequences have no predictable pattern. For example, if you rolled a die and recorded the numbers that came up, you might get:

$3, 1, 2, 2, 6, 4, 3, \ldots$

Because there is no pattern connecting these numbers, you cannot predict what the next term will be. Random sequences are not the focus of this topic.

Recursive sequences

Recursion involves creating sequences by repeatedly applying a rule to a starting number. These sequences have recognizable patterns that allow you to predict future terms. The rule tells you how to get from one term to the next.

For instance, if you start with $1$ and use the rule "add $2$ to each term," you generate the sequence:

$1, 3, 5, 7, 9, \ldots$

Knowing both the starting value and the rule means you can easily find any term in the sequence.

Behaviour of sequences

When sequences follow a pattern, they can exhibit different types of behaviour. Understanding these behaviours helps you classify and analyze sequences.

Increasing sequences

An increasing sequence is one where each term is larger than the previous term. The values get progressively bigger.

Example: $1, 4, 9, 25, 36, \ldots$

Each term increases from the one before it.

Decreasing sequences

A decreasing sequence is one where each term is smaller than the previous term. The values get progressively smaller.

Example: $100, 90, 80, 70, \ldots$

Each term decreases by $10$ from the previous term.

Constant sequences

A constant sequence is one where all terms have exactly the same value. The sequence doesn't change.

Example: $-3, -3, -3, -3, \ldots$

Every term equals $-3$.

Oscillating sequences

An oscillating sequence is one where the terms alternate or change between two or more values. The sequence bounces back and forth.

Example: $1, -1, 1, -1, \ldots$

The sequence alternates between $1$ and $-1$.

Another example: $10, -8, 10, -8, \ldots$

The sequence oscillates between $10$ and $-8$.

Limiting values

Some sequences have a limiting value, which means the terms approach a particular number as the sequence continues. The terms get closer and closer to this value without necessarily reaching it.

Example: $\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots$

As the sequence progresses, the terms get smaller and approach zero. We say this sequence has a limiting value of zero.

Another example: $1000, 100, 10, 1, 0.1, \ldots$

Each term is divided by $10$, so the sequence decreases and approaches zero as a limiting value.

Understanding recursion

Different types of rules can be used, including:

• Adding or subtracting a number
• Multiplying or dividing by a number
• Squaring numbers
• Combining previous terms

The key to recursion is that you need two pieces of information:

1. The starting value (first term)
2. The rule that tells you how to find the next term

Once you have these, generating the sequence is straightforward.

Finding recursive rules

To identify the pattern in a sequence, you need to examine how consecutive terms relate to each other. Look at the difference between terms or the ratio between them.

Tips for finding patterns

Generating sequences from starting values and rules

When you're given a starting value and a rule, you can generate as many terms as needed by repeatedly applying the rule.

Step-by-step process

When generating a sequence:

1. Write down the starting value as your first term
2. Apply the rule to the first term to get the second term
3. Apply the rule to the second term to get the third term
4. Continue applying the rule until you have the required number of terms
5. Write your answer as a sequence with terms separated by commas

Using calculators to generate sequences

While you can generate sequence terms by hand, using a CAS calculator makes the process much faster, especially when you need many terms. The calculator can automate the repetitive application of a recursive rule.

Calculator method for addition-based sequences

If your rule involves adding the same number repeatedly, you can use the calculator's "Answer" function to speed up calculations.

Benefits of calculator methods

Using a calculator for sequence generation:

• Saves time when finding many terms
• Reduces arithmetic errors
• Allows you to explore longer sequences
• Helps verify patterns you've identified by hand

Exam tips

Remember!