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:
Tn = 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 × 1.5n-1.
In general, for a geometric sequence with first term a and common ratio r, the nth term is given by:
Tn = arn-1
In both cases the rule for finding the n
th 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.