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.

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 (5x7)-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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