A **sequence** is a series of shapes or numbers which follow a rule.

An **arithmetic sequence** is a sequence of numbers with the rule ‘add a fixed number’. The number is the **difference** between consecutive terms.

For example, in the sequence 5, 8, 11, 14, 17, …, the rule is ‘add 3’. The nth term is 3n + 2. The 3 in 3n + 2 is the difference between consecutive terms. The +2 in 3n + 2 is the term that would come before the 1st term, sometimes called the zero term.

In general, for an arithmetic sequence with first term a and difference d, the nth term is given by:

T_{n} = a + (n – 1)d

A visual example of an arithmetic sequence:

*Write down the number of dots in pattern 4. Find an expression, in terms of n, for the nth term in the sequence.*

Does any pattern in the sequence contain 245 dots?

To create the next pattern requires adding 4 more dots, giving 19 dots, and the nth pattern will have 4n + 3 dots in it.

If a pattern has 245 dots then the equation 4n + 3 = 245 will have an integer solution. n =

= 60.5 shows that no pattern in the sequence will have 245 dots in it.

A **geometric sequence** is a sequence of numbers with the rule ‘multiply by a fixed number’. The number, called the **common ratio**, is the ratio between successive terms.

For example, in the sequence 24, 36, 54, 81, 121.5, …, the rule is ‘multiply by 1.5’. The nth term is 24 × 1.5^{n-1}.

In general, for a geometric sequence with first term a and common ratio r, the nth term is given by:

T_{n} = ar^{n-1}

In both cases the rule for finding the

nth term involves applying the term-to-term rule exactly

n – 1 times.

If the sequence is 2, 5, 8, 11

A finite series would be 2+5+8+11.

We could write the sum as

We can add up these terms to get 26 but if there were many more terms we would need a formula.

If an arithmetic series has first term a and common difference d, then the sum of the first n terms of the progression is

n(2a + ( n − 1 )d).

An understanding of sequences is an important step in generalising. Many using and applying problems can be generalised to an arithmetic sequence or a geometric sequence.

These basic sequences help to develop an understanding of the more complex sequences; e.g. consecutive squares found in A and AS level curriculum.