Nth Term | Mathematics for GCSE/IGCSE - Year 11 PDF Download

Linear Sequences

 What is a linear sequence? 

• A linear sequence is a sequence where the terms increase or decrease by the same amount each time. For example, 1, 4, 7, 10, 13, ... (add 3 to get the next term) or 15, 10, 5, 0, -5, ... (subtract 5 to get the next term).
• This type of sequence is also known as an arithmetic sequence. By observing the differences between consecutive terms, you will find that they remain constant.

What should we be able do with linear sequences?

• You should be able to recognize and extend a linear sequence.
• You should also be able to determine a formula for the n^th term of a linear sequence in relation to n.
• This formula is usually in the form: nth term = dn + b, where;
  • d represents the common difference.
  • b is a constant that adjusts the first term appropriately.

How do I find the nth term of a linear (arithmetic) sequence?

• Determine the common difference between the terms - this value is represented by d.
• Substitute the initial term and n = 1 into the formula, then solve for b.

Quadratic Sequences

What is a quadratic sequence?

• In a quadratic sequence, unlike in a linear one, the differences between consecutive terms (first differences) aren't constant. 
• However, the differences between these first differences (second differences) remain constant. 
• Another perspective is that the sequence of first differences forms a linear sequence in a quadratic sequence. For instance:
  • Sequence: 2, 3, 6, 11, 18, …
  • First Differences: 1, 3, 5, 7 (a Linear Sequence)
  • Second Differences: 2, 2, 2 (Constant)
• When the second differences are constant, we can identify the sequence as quadratic.

What should I be able to do with quadratic sequences?

• You should be able to recognize and continue a quadratic sequence.
• You should also be able to find a formula for the nth 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 nth term of a quadratic sequence?

• STEP 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,  ...
• STEP 2
  • Find the value of a using a = [the second difference] ÷ 2 
    • e.g.  a = 4 ÷ 2 = 2
• STEP 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, ...
• STEP 4
  • Work out the linear n^th term of these differences. This is bn + c.
    • e.g.  bn + c = 3n − 4
• STEP 5
  • Add this linear n^th term to an² 
  • Now you have an² + bn + c.
    • e.g.  an² + bn + c = 2n² + 3n − 4