**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

- Look for anything that makes the position-to-term and/or the term-to-term rule easy to spot

- 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}

is called the**r****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)

**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

- ie The term-to-term rule is
- 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

- 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

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, ...

b)

The 3rd and 6th terms in a Fibonacci sequence are 7 and 31 respectively.

Find the 1st and 2nd terms of the sequence.

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