HSC Maths Advanced Syllabus

What are Sequences and Series?

A sequence is an ordered list of numbers. Each number in the sequence is called a term. Sequences can be finite or infinite, and they are typically defined by a rule or formula that describes the relationship between the position of a term and its value.

A series is the sum of the terms of a sequence. If the sequence is infinite, the series is called an infinite series.

Infinite and Finite Sequences

An infinite sequence is indicated by arranging positive integers in ascending order. To show that the sequence goes on forever, three dots ... are written at the end of the sequence.

$$1 , 2, 3, 4, 5, ...$$

A finite sequence is indicated by listing all of the terms of the sequence up to the final term which acts as the definite end.

$$1, 2,3,4,5,6$$

To identify a specific term in a sequence we use the symbol $T_n$ whereby $n$ is the position of the term in the sequence. For the above sequences:

$$T_1 = 1\qquad T_2=2\qquad T_3=3$$

Sequences can be defined in three ways

Listing the Initial Terms: This involves writing out the start of the sequence to establish the pattern. Such as: $1, 2, 3, 4, 5, ...$

Providing a Formula for the nth Term: A formula offers a direct way to find any term in the sequence. For example, we can define the term of our sequence as $T_n = 5n - 2$, this rule goes on infinitely unless a constraint is placed upon it, as for any term $T$ at position $n$ we can calculate the value.

For example: Find the 5th term using the following formula: $T_n=5n-2$

$$T_5=5(5)-2 \\ = 25 -2 \\ =23$$

Using a Recursive Formula: This method defines each term based on the previous one(s).

For example, a sequence starts at 5, with each term progressively increasing in increments of 5. Consider the below formula, $T_n = T_{n-1} + 5$ given that we have stated when $n =1$, $T_1 = 5$. When $n = 2$, we can plug in $T_2 = T_{1} + 5$ which we can equate as $T_2 = (5) + 5$. Thus for all positions n, we can find the term $T_n$ so long as we compute every single term before $T_n$.

$$T_1=5\newline T_n=T_{n-1} +5\qquad (n\geq2)$$