Formula for `n^(th)` term of a sequence - consistent difference between differences

The formula is `an^2+bn+c` and this is explained by the following:

The following pattern is known as a quadratic.

 

When we know we are dealing with a sequence that is a quadratic (where the difference between differences are consistent) we must remember the facts about quadratics.

See mammoth memory quadratics to understand the above picture.

(A quadratic is an `x^2`  term `=ax^2+bx+c`)

 

A quadratic is

`ax^2+bx+c`

but here the `x`  is replaced by the letter `n`  denoting the number in a sequence.

The quadratic becomes

 

and this is all you need to remember because

 

The first term is

1st term (where `n=1`) `=atimes1^2+btimes1+c`

is the same as

1st term`=a+b+c`

 

The second term is

2nd term `(n=2)=atimes2^2+btimes2+c`

is the same as

2nd term `=4a+2b+c`

 

The third term is

3rd term `(n=3)=atimes3^2+btimes3+c`

is the same as

3rd term `=9a+3b+c`

 

Summary

So we know

So for any quadratic sequence (consistent difference between differences) we can use this information to work out ANY `n^(th)`  term formula.

 

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