Recursion With Geometric Sequences Revision Notes for VCE SSCE General Mathematics

Recursion With Geometric Sequences

What is a recurrence relation?

A recurrence relation is a mathematical way to describe how a sequence is generated. It tells you two things:

Where to start (the initial value)
How to get from one term to the next (the rule)

Instead of listing all the terms, a recurrence relation gives you a precise mathematical formula that captures the pattern.

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Understanding Recurrence Relations

A recurrence relation is powerful because it provides a complete description of a sequence using just two pieces of information: a starting point and a rule for progression. This compact representation can describe infinitely many terms.

Example: The sequence 2, 6, 18, ...

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Worked Example: Describing a Sequence with a Recurrence Relation

We can describe this sequence in words:

"Start with 2"
"To find the next term, multiply the current term by 3, and keep repeating"

Using mathematical notation with terms labeled $t_0, t_1, t_2, \ldots$ :

$t_0 = 2$ (initial or starting value)
$t_1 = 3 \times t_0 = 6$ (after 1 application of the rule)
$t_2 = 3 \times t_1 = 18$ (after 2 applications of the rule)
$t_3 = 3 \times t_2 = 54$ (after 3 applications of the rule)
$t_4 = 3 \times t_3 = 162$ (after 4 applications of the rule)

And generally, $t_{n+1} = 3 \times t_n$ (after $n$ applications of the rule).

The recurrence relation for this sequence is written as:

$t_0 = 2, \quad t_{n+1} = 3t_n \quad \text{for } n = 0, 1, 2, 3, \ldots$

This says: "The first term is 2, and each subsequent term equals the current term multiplied by 3."

General recurrence relation for geometric sequences

For any geometric sequence, the recurrence relation has the form:

$t_0 = a, \quad t_{n+1} = Rt_n$

Where:

$a$ is the starting value (the first term, $t_0$ )
$R$ is the common ratio (the number you multiply by each time)

This generates a geometric sequence where each term is $R$ times the previous term.

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The General Form

Every geometric sequence can be expressed using this standard recurrence relation form: $t_0 = a, \; t_{n+1} = Rt_n$ . Recognizing this pattern is essential for working with geometric sequences efficiently.

Generating a geometric sequence using recursion

Let's work through an example to see how this works in practice.

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Worked Example: Generating the First Five Terms

Problem: Generate and graph the first five terms of the sequence defined by:

$t_0 = 5, \quad t_{n+1} = 2t_n$

Solution:

Step 1: Identify the starting value

$t_0 = 5$

Step 2: Apply the rule repeatedly

The rule says "to find the next term, multiply the previous term by 2":

$t_1 = 2t_0 = 2 \times 5 = 10$
$t_2 = 2t_1 = 2 \times 10 = 20$
$t_3 = 2t_2 = 2 \times 20 = 40$
$t_4 = 2t_3 = 2 \times 40 = 80$

Step 3: Graph the terms

To graph, we plot $t_n$ against $n$ for $0 \leq n \leq 4$ :

The graph shows the exponential growth of the geometric sequence.

Finding the nth term directly

While repeated multiplication works, it becomes very tedious for finding terms like $t_{50}$ . Instead, we can develop a direct formula to calculate any term without finding all the previous ones.

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Why We Need a Direct Formula

Using the recurrence relation to find $t_{50}$ would require calculating all 50 previous terms one by one. The direct formula allows us to jump straight to any term, making calculations much more efficient.

Developing the pattern

Consider the geometric sequence: $1, 4, 16, 64, \ldots$ defined by:

$t_0 = 1, \quad t_{n+1} = 4t_n \quad \text{for } n = 0, 1, 2, 3, \ldots$

Let's trace what happens at each step:

$t_0 = 1$ (after 0 applications of the rule)
$t_1 = 4 \times t_0 = 4^1 \times t_0 = 4$ (after 1 application)
$t_2 = 4 \times 4 \times t_0 = 4^2 \times t_0 = 16$ (after 2 applications)
$t_3 = 4 \times 4 \times 4 \times t_0 = 4^3 \times t_0 = 64$ (after 3 applications)

A pattern emerges: after $n$ applications of the rule:

$t_n = 4^n \times t_0$

Using this formula, we can find any term directly. For example:

$t_{10} = 4^{10} \times 1 = 1\,048\,576$

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Recognizing the Pattern

Notice how the number of multiplications becomes the exponent in the direct formula. After applying the rule $n$ times, we've multiplied by the common ratio $n$ times, which is equivalent to raising it to the power of $n$ .

General formula for the nth term

This pattern can be generalised for any geometric sequence.

If the recurrence relation is:

$t_0 = a, \quad t_{n+1} = R \times t_n$

Then the nth term can be found directly using:

$t_n = R^n \times a$

Where:

$n$ is the term number ( $n = 0, 1, 2, 3, \ldots$ )
$a$ is the starting value
$R$ is the common ratio

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The Direct Formula

The formula $t_n = R^n \times a$ is one of the most important results for geometric sequences. Memorize this formula and understand that:

The exponent $n$ represents how many times you've applied the rule
You multiply by the starting value $a$ at the end
This works for any geometric sequence, no matter how large $n$ is

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Worked Example: Finding a Specific Term

Problem: Consider the recurrence relation $t_0 = 2, \; t_{n+1} = 3t_n$ . Find $t_{12}$ .

Solution:

Step 1: Identify the starting value and common ratio

$a = 2$
$R = 3$

Step 2: Identify the term number

We want $t_{12}$ , so $n = 12$ (this requires 12 applications of the rule)

Step 3: Substitute into the formula $t_n = R^n \times a$

$t_n = R^n \times a$

$t_{12} = 3^{12} \times 2$

Step 4: Evaluate

$t_{12} = 531\,441 \times 2 = 1\,062\,882$

Answer: $t_{12} = 1\,062\,882$

Key formulas summary

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Essential Formulas for Geometric Sequences

For a geometric sequence with starting value $a$ and common ratio $R$ :

Recurrence relation (recursive definition):

$t_0 = a, \quad t_{n+1} = R \times t_n$

This tells you how to generate the sequence step by step.

Direct formula (explicit definition):

$t_n = R^n \times a$

This allows you to find any term directly without calculating all previous terms.

Both formulas describe the same geometric sequence, but the direct formula is much more efficient for finding terms far along in the sequence.

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Key Points to Remember:

A recurrence relation specifies a starting value and a rule for generating subsequent terms
For geometric sequences, the recurrence relation is $t_0 = a, \; t_{n+1} = Rt_n$
The direct formula $t_n = R^n \times a$ lets you find any term without calculating all previous ones
In the formula, $n$ represents the number of times the rule has been applied (term number starting from 0)
The direct formula is derived by recognising the pattern of repeated multiplications

Recursion With Geometric Sequences (VCE SSCE General Mathematics): Revision Notes