Sequences and How to Specify Them (HSC SSCE Mathematics Advanced): Revision Notes

Sequences and How to Specify Them

What is a sequence?

A sequence is an ordered list of numbers that follows a specific pattern or rule. Sequences can be either infinite (continuing forever) or finite (ending after a certain number of terms).

Infinite sequence example:

The positive odd integers arranged in increasing order form an infinite sequence:

$1, 3, 5, 7, 9, 11, 13, 15, 17, 19, \ldots$

The three dots ($\ldots$) indicate that the sequence continues forever with no last term.

Finite sequence example:

The two-digit odd numbers less than $100$ form a finite sequence:

$11, 13, 15, 17, \ldots, 99$

Here, the dots represent the $45$ terms that have been left out for brevity.

Understanding sequence notation

In mathematics, we use special notation to describe sequences clearly:

• Tₙ represents the nth term of a sequence
• T₁ is the first term
• T₂ is the second term
• T₃ is the third term, and so on

For example, in the sequence $1, 3, 5, 7, 9, \ldots$:

$T_1 = 1, \quad T_2 = 3, \quad T_3 = 5, \quad T_4 = 7, \quad T_5 = 9, \ldots$

Three ways to specify a sequence

There are three different methods you can use to define or describe a sequence. Understanding all three methods will help you work flexibly with sequences.

Method 1: Write out the first few terms

This is the simplest approach. You write out enough terms until the pattern becomes obvious to anyone reading it.

Example:

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

From this, we can see the pattern continues as $11, 13, 15, 17, 19, \ldots$

With some calculation, you can work out that:

• $T_{11} = 21$
• $T_{14} = 27$
• $T_{16} = 31$

Method 2: Give a formula for the nth term

A formula provides a direct way to calculate any term in the sequence without needing to know previous terms.

Example:

For the sequence of positive odd integers, the formula is:

$T_n = 2n - 1$

This works because the nth odd number is always $1$ less than $2n$.

Using this formula, we can quickly calculate any term:

$T_{30} = 60 - 1 = 59$

$T_{100} = 200 - 1 = 199$

$T_{244} = 488 - 1 = 487$

Method 3: Say where to start and how to proceed

This method, called a recursive formula, describes:

1. The starting value (first term)
2. How to get from one term to the next

Example:

For the positive odd integers:

• $T_1 = 1$ (start with $1$)
• $T_n = T_{n-1} + 2$, for $n \geq 2$ (each term is $2$ more than the previous term)

Using the formula for Tₙ to solve problems

Many problems involving sequences can be solved by setting up and solving an equation using the formula for $T_n$. This section demonstrates how to apply sequence formulas to various types of problems.