Types-Of-Sequences Aqa Higher

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

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

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