Sequences (CIE IGCSE Maths: Extended)

What are sequences?

• A sequence is an ordered set of (usually) numbers
• Each number in a sequence is called a term
• The location of a term within a sequence is called its position
  • The letter n is often used for (an unknown) position

• Subscript notation is used to talk about a particular term
  • a₁ would be the first term in a sequence
  • a₇ would be the seventh term
  • a_n would be the n^th term

What is a position-to-term rule?

• A position-to-term rule gives the n^th term of a sequence in terms of n
  • This is a very powerful piece of mathematics
  • With a position-to-term rule the 100^th term of a sequence can be found without having to know or work out the first 99 terms!

What is a term-to-term rule?

• A term-to-term rule gives the (n+1)^th term in terms of the n^th term
  • ie a_n+1 is given in terms of a_n
  • If a term is known, the next one can be worked out

How do I use the position-to-term and term-to-term rules?

• These can be used to generate a sequence
• From a given sequence the rules can be deduced
• Recognising and being aware of the types of sequences helps
  • Linear and quadratic sequences
  • Geometric sequences
  • Fibonacci sequences
  • Other sequences

What other 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