nth Term (CIE IGCSE Maths: Core)

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

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 I 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 simple quadratic sequence in terms of n
  • This formula will be in one of the forms:

    n^th term = an² 
    n^th term = (n+b)² 
    n^th term = n² + b
    n^th term = an² + b
    

• To find the values of a and b, you must remember the terms in the sequence for n² and compare them to the given sequence
  • n² is the sequence 1, 4, 9, 16, 25, 36, 49, ....
  • Usually, each term will be either a little bit more or less than the sequence for n²
    • For example, the sequence 2, 5, 10, 17, 26, 37, 50, ... has the formula n² + 1 as each term is 1 more than the corresponding term in n²
  • Sometimes, each term will be two or three times (or maybe even one half of) the term in the sequence for n²
    • For example, the sequence 2, 8, 18, 32, 50, 72, 98, ... has the formula 2n²  as each term is twice the corresponding term in n²

What is a cubic sequence?

• In a cubic sequence the differences between the terms (the first differences) are not constant and the differences between the differences (the second differences) are not constant
• However, the differences between the second differences (the third differences) are constant
• Another way to think about this is that in a cubic sequence, the sequence of second differences is a linear sequence

  eg Sequence:   1, 5, 21, 55, 113, …

  1st Differences:  4,  16,  34,  58 (a Quadratic Sequence)

  2nd Differences:   12,  18,  24 (a Linear Sequence)
  3rd Differences:  6,  6,  6 (Constant)

• If the third differences there are constant, we know that the example is a cubic sequence

What should I be able to do with cubic sequences?

• You should be able to recognise and continue a cubic sequence
• You should also be able to find a formula for the n^th term of a simple cubic sequence in terms of n
  • This formula will be in one of the forms:

    n^th term = an³ 
    n^th term = n³ + b
    

• To find the values of a and b, you must remember the terms in the sequence for n³ and compare them to the given sequence
  • n³ is the sequence 1, 8, 27, 64, 125, ....
  • Usually, each term will be either a little bit more or less than the sequence for n³
    • For example, the sequence 2, 9, 28, 65, 126, , ... has the formula n³ + 1 as each term is 1 more than the corresponding term in n³
  • Sometimes, each term will be two or three times the term in the sequence for n³
    • For example, the sequence 2, 16, 54, 128, 250, ... has the formula 2n³  as each term is twice the corresponding term in n³