Home >> Number - Sequences

**Coursework Notes - Number**

**Sequences**

[ common difference sequences ][changing difference sequences]

Conventionally sequences have a **first term** or starting value, usually denoted by the letter '**a**'.

The **common difference** '**d**' is the difference between consecutive terms when the terms increase by a regular amount.

**The difference change** '**c**' is the change between consecutive differences

The **last term** in a sequence of '**n**' numbers is the **nth** term.

The **general term** is an expression in 'n' that can be used to calculate any term in the sequence.

__'Common Difference' Sequences __

The __general term__ for term number '**n**' , common diff. '**d**' and first term '**a**' is:

**dn + (a - d) **

e.g. : 4.....9.....14.....19.....24.....29.....

a = 4, d = 5

the nth term is **dn + (a - d)** = 5n + (4 - 5) = 5n - 1

n = 7 , 7th term is (5 x 7) - 1 = 34

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__example #1__ - Find the nth term in this sequence : 13, 20, 27, 34, 41, 48 ...

a = 13, d = 7

nth term = dn + (a - d) = 7n + (13 - 7) = __7n +6 __

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__example #2__ - Find the nth term in this sequence : 11, 19, 27, 35, 43, 51 ...

a = 11, d = 8

nth term = dn + (a - d) = 8n + (11 - 8) = __8n + 3 __

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__example #3__ - Find the nth term in this sequence : 9, 15, 21, 27, 33, 39 ...

a= 9, d= 6

nth term = dn + (a - d) = 6n + (9 - 6) = __6n + 3__

__'Changing Difference' Sequences __

The __general term__ for term number '**n**' , common diff. '**d**' , first term '**a**' and difference change '**c**' is:

__Example #1__ - find the nth term of 3, 8, 14, 21, 29 ........

Writing the series with increases below:

remembering that the nth term is given by:

1st term, 'a' =3

first difference 'd' = 5

difference increase 'c' = 1

__Example #2__ - find the nth term of 5, 7, 10, 14, 19 ........

Writing the series with increases below:

remembering that the nth term is given by:

1st term, 'a' =5

first difference 'd' = 2

difference increase 'c' = 1

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