Nth term sequences

Arithmetic sequences (common difference)

… When the first difference is constant, e.g. +3 each time, then the nth term will have a 3n in the expression. Test this out and then make a modification if you need to by adding or subtracting a number.

Geometric sequences (common ratio)

… Instead of a common difference between each term, common ratio sequences (called geometric sequences) have the same multiplier between each term. For example:

20, 10, 5, 2.5, …

Here, the common ratio (multiplier) is…?

You don’t need to know the formula for these types of sequences – just spot the pattern and see if you can work out the multiplier.

Term-to-term sequences

… The next term in a “term-to-term” sequence depends on the previous term!

Here’s a term-to-term rule:

U[n+1] = 2U[n] – 3 [the square brackets are written in subscript]

This means, “The next term = (2 x previous term) – 3”

So, if U[1] = 4

U[2] = 2U[1] – 3 = 2(4) – 3 = 5

U[3] = 2U[2] – 3 = 2(5) – 3 = 7

and so on…

Quadratic sequences

… When the second difference is constant you have a QUADRATIC SEQUENCE! Follow this four step process:

1) Write out the sequence showing the first difference and the second difference

2) Working diagonally downwards to the right, circle:

the first term,

the first difference (just below),

and the second difference (just below).

3) Write down the following (you’ll need to remember this!):

a + b + c = first term

3a + b = first difference

2a = second difference

4) Use the equations you have formed to work out a, then b and then c.

5) your nth term expression (Un means nth term) will be:

Un = an² + bn + c

Test this to check that the expression works for the sequence.

Here are two examples of quadratic sequences. See if you can use the above method to find an expression for the nth term:

a) 0 , 3 , 10 , 21 , 36

b) -2 , 11 , 32 , 61 , 98

Triangular numbers

… This sequence is based on triangles. Try constructing triangles using dots and you’ll get the sequence:

1, 3, 6, 10…

Notice that you’re adding 1 more each time, which actually forms a quadratic sequence.

Fibonacci sequence

Add the previous two terms together to make the next term:

0, 1, 1, 2, 3, 5, 8…

Fibonacci sequences often occur in nature, for example in the spiral of a sea shell.