Types of Sequences (Cambridge O Level Maths)

What types of sequences are there?

• Linear and quadratic sequences are particular types of sequence covered their own notes
• Other sequences include geometric and Fibonacci sequences, which are looked at in more detail below
• Other sequences include cube numbers (cubic sequences) and triangular numbers
• Another common type of sequence in exam questions, is fractions with combinations of the above
  • Look for anything that makes the position-to-term and/or the term-to-term rule easy to spot

What is a geometric sequence? 

• A geometric sequence can also be referred to as a geometric progression and sometimes as an exponential sequence
• In a geometric sequence, the term-to-term rule would be to multiply by a constant, r
  • a_n+1 = r.a_n
• r is called the common ratio and can be found by dividing any two consecutive terms, or
  • r = a_n+1 / a_n
• In the sequence 4, 8,  16,  32,  64, ... the common ratio, r,  would be 2 (8 ÷ 4 or 16 ÷ 8 or 32 ÷ 16 and so on)

What is a Fibonacci sequence? 

• THE Fibonacci sequence is 1, 1,  2,  3,  5,  8,  13,  21,  34,  55, ...
• The sequence starts with the first two terms as 1
• Each subsequent term is the sum of the previous two
  • ie The term-to-term rule is a_n+2 = a_n+1 + a_n
  • Notice that two terms are needed to start a Fibonacci sequence

• Any sequence that has the term-to-term rule of adding the previous two terms is called a Fibonacci sequence but the first two terms will not both be 1
• Fibonacci sequences occur a lot in nature such as the number of petals of flowers

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 are constant, we know that the example is a cubic sequence

What should I be able to do with cubic sequence?

• 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 most likely be in one of the forms n^th term = an³ or 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

Problem solving and sequences

• When the type of sequence is known it is possible to find unknown terms within the sequence
• This can lead to problems involving setting up and solving equations
  • Possibly simultaneous equations

• Other problems may involve sequences that are related to common number sequences such as square numbers, cube numbers and triangular numbers

How do I identify a sequence?

• Is it obvious?
• Does it tell you in the question?
• Is there is a number that you multiply to get from one term to the next?
  • If so then it is a geometric sequence

• Next, look at the differences between the terms
  If 1^st differences are constant – it is a linear sequence

  If 2^nd differences are constant – it is a quadratic sequence

• Special cases to be aware of:
• If the differences repeat the original sequence
  • It is a geometric sequence with common ratio 2

• Fibonacci sequences also have differences that repeat the original sequence
  • However questions usually indicate if a Fibonacci sequence is involved

How do I continue a given sequence? 

• You may be given the first few terms of a sequence and be asked to find the next few terms
  • The key is to recognise the type of sequence and use the term to term rule to continue the sequence
• CASE 1: Look first to see if the sequence has a common difference
  • If it does then it is a linear sequence
    • Find the next terms by adding or subtracting that common difference to each term in turn
    • This is an example of using the term to term rule
• CASE 2: If the sequence does not have a common difference then look to see if it has a common second or third difference
  • If it has a common second difference then it is a quadratic sequence 
  • If it has a common third difference then it is a cubic sequence
    • Find the next term by first finding the next difference and then adding this to the sequence
• CASE 3: If there is no second or third common difference then look to see if it has a common multiplier (ratio)
  • If it does then it is a geometric sequence
    • Find the next term by multiplying the last term by the common multiplier
  • Look to see if there is a common value you can multiply each term by to get to the next term
• CASE 4: Some sequences may not fit into any of these categories, in this case you should look to see if there is a pattern in the difference which could help you to find the next term
  • For example the differences may form a geometric sequence which can be used to find the next difference and hence, the next value in the sequence
• You may also be asked to fill in some gaps within a sequence
  • In this case you will need to use the information you have to determine the type of sequence and to fill in the gaps