Writing Recurrence Relations in Symbolic Form Revision Notes for VCE SSCE General Mathematics

Writing Recurrence Relations in Symbolic Form

In mathematics, we need a clear and efficient way to describe sequences. This note explains how to use symbolic notation to label terms in a sequence and how to express the rules that generate sequences using recurrence relations.

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Symbolic notation provides a compact and precise method for representing sequences, making it easier to communicate mathematical ideas and perform calculations efficiently.

Numbering and naming the terms in a sequence

When working with sequences, we label each term using symbols with subscripts. These labels make it easy to refer to specific terms in the sequence.

Subscript notation

We use symbols such as $t_0$ , $t_1$ , $t_2$ , and so on, to name the terms in a sequence. The numbers written as subscripts (0, 1, 2, etc.) tell us important information:

The subscript indicates how many times we have applied the generating rule
$t_0$ is called the starting term or initial term because it marks the beginning of the sequence
Each subsequent term has a subscript one larger than the previous term

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Common Mistake to Avoid:

The subscript notation $t_3$ is completely different from raising a term to a power, such as $t^3 = t \times t \times t$ . The subscript is simply a label, not an operation.

Always remember: subscripts name the terms, they don't indicate multiplication!

Example of term labelling

For the sequence: $7, 3, 4, 11, 15, 24, \ldots$

Term number ( $n$ )	0	1	2	3	$\ldots$
Term symbol	$t_0 = 7$	$t_1 = 3$	$t_2 = 4$	$t_3 = 11$	$\ldots$
Term name	term 0	term 1	term 2	term 3	$\ldots$

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Worked Example: Naming terms in a sequence

Question: For the sequence: $2, 8, 14, 20, 26, 32, \ldots$ , state the values of:

a $t_1$
b $t_4$
c $t_5$

Solution:

First, write the term label under each value in the sequence:

$2, \quad 8, \quad 14, \quad 20, \quad 26, \quad 32$

$t_0 \quad t_1 \quad t_2 \quad t_3 \quad t_4 \quad t_5$

Now read off the required values:

a $t_1 = 8$

b $t_4 = 26$

c $t_5 = 32$

Exam tip: Always remember that the first term of the sequence is $t_0$ , not $t_1$ . This is a common mistake to avoid.

Recurrence relations

A recurrence relation provides a mathematical method for generating the terms of a sequence. Understanding recurrence relations is essential for working with sequences efficiently.

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Two Essential Components:

A recurrence relation consists of two parts that must always work together:

A starting point: the value of the first term in the sequence
A rule: instructions for generating each successive term from the previous term

Both parts are necessary—without either one, you cannot fully define the sequence!

Converting descriptions to symbolic form

Consider the sequence: $2, 8, 14, 20, \ldots$

We could describe this sequence in words as:

Start with 2
To obtain each new term, add 6 to the current term, then repeat the process

However, we can express this much more efficiently using symbolic notation. Let $t_n$ represent the term after $n$ applications of the rule. The recurrence relation can then be written compactly as:

The table shows how the three components relate to each other:

The starting value tells us where to begin: $t_0 = 2$
The rule shows how to find the next term: $t_{n+1} = t_n + 6$
The complete recurrence relation combines both: $t_0 = 2, \; t_{n+1} = t_n + 6$

Understanding the notation

In the expression $t_{n+1} = t_n + 6$ :

$t_n$ represents the current term
$t_{n+1}$ represents the next term
The equation tells us: "the next term equals the current term plus 6"

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Flexibility in Notation:

The letter $t$ can be replaced by any letter of the alphabet (such as $a$ , $u$ , or $s$ ). The choice of letter doesn't change the mathematical meaning.

Each application of the rule is called an iteration. This term is used throughout mathematics and computer science to describe repeated processes.

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Worked Example: Generating a sequence from a recurrence relation

Question: Write down the first five terms of the sequence defined by the recurrence relation:

$t_0 = 30, \quad t_{n+1} = t_n - 5$

Solution:

Start with the initial value:

$t_0 = 30$

Apply the rule to find the first term:

$t_1 = 30 - 5 = 25$

Continue applying the rule to find the remaining terms:

\begin{aligned} t_2 &= t_1 - 5 = 25 - 5 = 20 \\ t_3 &= t_2 - 5 = 20 - 5 = 15 \\ t_4 &= t_3 - 5 = 15 - 5 = 10 \end{aligned}

Therefore, the first five terms are:

$30, \; 25, \; 20, \; 15, \; 10$

Exam tip: When generating terms from a recurrence relation, write out each step clearly. This helps avoid calculation errors and makes your working easy for examiners to follow.

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Key Points to Remember:

Subscript notation uses symbols like $t_0$ , $t_1$ , $t_2$ to label terms in a sequence, where the subscript indicates how many times the rule has been applied
The sequence always starts with $t_0$ , which is called the starting term or initial term
A recurrence relation has two parts: a starting value and a rule for generating successive terms
The symbolic form $t_0 = a, \; t_{n+1} = t_n + d$ provides a compact way to describe an entire sequence
Each application of the rule in a recurrence relation is called an iteration
When using recurrence relations, always identify the starting value first, then apply the rule repeatedly to generate the sequence

Writing Recurrence Relations in Symbolic Form (VCE SSCE General Mathematics): Revision Notes