Types-Of-Sequences Aqa Foundation

Types of sequences

What types of sequences are there?

• Linear sequences are the sequences that you are most likely to see in an exam question

  • These are sometimes called arithmetic sequences or progressions

  • A common difference is added to or subtracted from one term to get to the next term

• Other types of sequences that you may also come across include

  • Quadratic sequences (square numbers)

  • Cube numbers

  • Triangular numbers

  • Geometric sequences

  • Fibonacci sequences

• 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 quadratic sequence?

• A quadratic sequence is based around square numbers

• The second differences are constant (the same)

  • Second differences are the differences between the first differences

  • For example, 2, 3, 6, 11, 18, …
    1st Differences: 1 3 5 7

    2nd Differences: 2 2 2

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

• This multiplier is called the common ratio and can be found by dividing any two consecutive terms

  • Consider the sequence 4, 8, 16, 32, 64, …
    The common ratio would be x2 (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

  • 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

Problem solving with sequences

• When the type of sequence is known it is possible to find unknown terms within the sequence

• Possibly simultaneous equationsThis can lead to problems involving setting up and solving equations

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