The general form of a quadratic sequence is given by

$$U_n=a{n}^2+bn+c$$

The first term is given by

$$U_1=a+b+c$$

The difference between each term is

$$U_{n+1}-U_n=a{(n+1)}^2+b(n+1)+c-(a{n}^2+bn+c)$$ $$=a(n^2+2n+1)+bn+b+c-a{n}^2-bn-c$$ $$=a{n}^2+2an+a+b-a{n}^2$$ $$=2an+a+b$$

So the difference between terms is a quadratic sequence is

$$U_{n+1}-U_n=2an+a+b$$

& the first difference between terms is

$$U_2-U_1=3a+b$$

Now taking the difference of the differences between terms, or, sometimes called, the second difference, that is, a difference minus the previous difference,

$$(U_{n+2}-U_{n+1})-(U_{n+1}-U_n)$$ $$=(2a(n+1)+a+b)-(2an+a+b)$$ $$=2a$$

So the difference between the differences in terms (or the second difference) is the constant $2a$.

We can use these results to help determine the values of a, b & c.