Types of Sequences (OCR GCSE Maths)

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Jamie W

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Jamie W

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Types of Sequences

What other sequences are there?

  • Linear and quadratic sequences are particular types of sequence covered in previous notes
  • Other sequences include geometric and Fibonacci sequences, which are looked at in more detail below
  • Other sequences include cube numbers 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

SeqOth Notes fig2, downloadable IGCSE & GCSE Maths revision notes

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
    • an+1 = r.an
  • r is called the common ratio and can be found by dividing any two consecutive terms, or
    • r = an+1 / an
  • 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) SeqOth Notes fig3, downloadable IGCSE & GCSE Maths revision notes

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 an+2 = an+1 + an
    • 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

 

SeqOth Notes fig4, downloadable IGCSE & GCSE Maths revision notes

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

 SeqOth Notes fig5, downloadable IGCSE & GCSE Maths revision notes

Worked example

a)
Identify the types of sequence below;
i)       4, 5, 9, 14, 23, 37, 60, ...
ii)      6, 10, 16, 24, 34, ...
iii)      12, 7, 2, -3, ...

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b)
The 3rd and 6th terms in a Fibonacci sequence are 7 and 31 respectively.
Find the 1st and 2nd terms of the sequence.

SeqOth Example fig2 solb, downloadable IGCSE & GCSE Maths revision notes

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Jamie W

Author: Jamie W

Expertise: Maths

Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.