Q.Construct the forward difference table for the data $x=0,1,2,3$; $y=2,5,10,17$, and find $\Delta^2y_0$.

The same repeated-differencing procedure applies to any table of equally-spaced data.

$\Delta y=3,5,7$; $\Delta^2y=2,2$.

First differences

Second differences

Constant second differences ($=2$) confirm this data fits a quadratic exactly.

Check (independent recomputation via the underlying function $y=x^2+2x+2$): $f(0)=2,f(1)=1+2+2=5,f(2)=4+4+2=10,f(3)=9+6+2=17$ — all match; the leading coefficient $1$ with $h=1$ gives the theoretical constant 2nd difference $2!\times1\times1=2$, matching exactly.

Stopping at the first differences and reporting $\Delta y_0=3$ when asked for $\Delta^2y_0$ — always re-read which ORDER of difference the question asks for before reporting a value.