SOLUTION: What is the nth term rule of the quadratic sequence below? 6 , 12 , 20 , 30 , 42 , 56 , 72 , . . .

The good lady mistakenly thought it was an arithmetic 
sequence, but it is a quadratic sequence, so we assume:



[Don't confuse the capital A with the subscripted small an,
for the nth term.  They are different.]

There are 3 unknowns A,B, and C, so we need 3 equations,
so we only need the first 3 terms, with n=1,2,3.

, , 

Substitute n=1:





So the first equation is 

Substitute n=2:





So the second equation is 

Substitute n=3:





So the third equation is 

We have this 3×3 system:



Subtract the first equation from the second:



Subtract the second equation from the third equation:



So we have this 2×2 system:



Subtract the first equation from the second:





Substitute in






Substitute in 






We substitute in the expression for the nth term 






    

If you like you can factor that as

    

Checking to make sure it works for n=1,...,7 with the given sequence:

 
 
 
 
 
 
 

Edwin