Q.Using Newton's backward interpolation formula on $x=1,2,3,4,5$; $y=2,5,10,17,26$, estimate $y$ at $x=4.5$.

Reusing the backward differences already computed for this table's last row, apply Newton's backward formula for a point near the end.

Setting up $p$

$x_5=5,h=1$, so $p=\dfrac{4.5-5}{1}=-0.5$.

Using the backward differences from the earlier worked example's table ($\nabla y_5=9$, $\nabla^2y_5=2$)

$=26-4.5+\frac{-0.25}{2}(2)=26-4.5-0.25=21.25$ …

Recomputing the backward differences from scratch instead of reusing the already-correct difference table from the earlier example on the same data — always check whether the requ …