Quadratic number pattern equation

Question

May I know how do I form a quadratic number pattern equation? I cant seem to form one on my own.

1500, 1519,1536, 1551,1564.

Answer 1

Compute the difference of successive terms, do it again, deduce that there is indeed a quadratic equation given by : $$t_n=1500+19\,\frac{n}{1!}-2\,\frac{n\,(n-1)}{2!},\quad n\ge 0$$

(the coefficients are given by $\ c_0=t_0,\ c_1=t_1-t_0,\ c_2=(t_2-t_1)-(t_1-t_0),\ c_3=0,\cdots$)

If you want only a quadratic fit (an approximative quadratic solution) you may try for example alpha to get the solution (with perfect fit $1$ here) : $$t(n)=-n^2+20\,n+1500$$

If you want to start at $n=1$ instead of $n=0\,$ replace $\,n\,$ by $\,n-1$.