Mammoth Memory

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.

This sequence is 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.

You need to revisit quadratic equations to understand the formula for the nth term

See mammoth memory quadratics to understand the above picture.

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


A quadratic is


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

The quadratic becomes

The nth term formula is an2+bn+c 

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`



So we know

Summary of 1st, 2nd and 3rd quadratic terms

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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