Finding nth Terms of Sequences (AQA GCSE Further Maths)

Linear Sequences

What is a linear sequence?

• A linear sequence is one where the terms go up (or down) by the same amount each time

  • eg 1, 4, 7, 10, 13, … (add 3 to get the next term)

  • 15, 10, 5, 0, -5, … (subtract 5 to get the next term)

• A linear sequence is often referred to as an arithmetic sequence

• If we look at the differences between the terms, we see that they are constant

What should we be able do with linear sequences?

• You should be able to recognise and continue a linear sequence

• You should also be able to find a formula for the n^th term of a linear sequence in terms of n

• This formula will be in the form:

  n^th term = dn + b, where;

• d is the common difference

• b is a constant that makes the first term “work”

How do I find the n^th term of a linear (arithmetic) sequence?

• Find the common difference between the terms – this is d

• Put the first term and n = 1 into the formula, then solve to find b

Quadratic Sequences

What is a quadratic sequence?

• Unlike in a linear sequence, in a quadratic sequence the differences between the terms (the first differences) are not constant

• However, the differences between the differences (the second differences) are constant

• Another way to think about this is that in a quadratic sequence, the sequence of first differences is a linear sequence

  eg Sequence:   2, 3, 6, 11, 18, …

  1st Differences:  1  3  5  7 (a Linear Sequence)

  2nd Differences:   2  2  2 (Constant)

• If the second differences there are constant, we know that the example is a quadratic sequence

What should we be able to do with quadratic sequences?

• You should be able to recognise and continue a quadratic sequence

• You should also be able to find a formula for the n^th term of a quadratic sequence in terms of n

• This formula will be in the form:

  n^th term = an² + bn + c

  (The process for finding a, b, and c is given below)

How do I find the n^th term of a quadratic sequence?

1. Work out the sequences of first and second differences

   Note: check that the first differences are not constant and the second differences are constant, to make sure you have a quadratic sequence!

   • e.g.              sequence:    1,    10,    23,    40,    61

     •       first difference:    9,     13,    17,    21, ...

     • second differences:      4,       4,       4,  ...

2. a = [the second difference] ÷ 2 

   • e.g.  a = 4 ÷ 2 = 2

3. Write out the first three or four terms of an² with the first three or four terms of the given sequence underneath.
   Work out the difference between each term of an² and the corresponding term of the given sequence.

   • e.g.  an² = 2n² = 2,   8,   18,   32, ...
          sequence =    1,  10,   23,  40, ...
          difference =  -1,   2,     5,     8, ...

4. Work out the linear nth term of these differences. This is bn + c.

   • e.g.  bn + c = 3n − 4

5. Add this linear nth term to an². Now you have an² + bn + c.

   • e.g.  an² + bn + c = 2n² + 3n − 4