I am trying exercises of the book mentioned in the title and I need help in the question no. 6.
Image of question ->
I tried drawing graph of both the areas, $ S_r$ and $S_{r-1}$, but I don't think I am proceeding in the right direction because then I have to subtract something which is not in the equation I have to subtract. So, I am doing it wrong.
Please give some hints.
The equation in question $6a$ explains which are the lattice points that differ between the regions $T_{r-1}$ and $T_{r}$. The region $T_{r-1}$ can be visualized by starting from the square that includes all $(r-1)^2$ points with $1\leq x \leq r-1$ and $1\leq y \leq r-1$, then dividing the square in two right triangles by an ideal diagonal traced from the upper left to the lower right vertex of the square, and finally taking the upper right triangle (including the diagonal). In this way, $T_{r-1}$ includes $\sum_{k=1}^{r-1} k$$=r(r-1)/2$ lattice points.
When we pass from $T_{r-1}$ to $T_{r}$, our triangular region gains the $r^{th}$ row and the $r^{th}$ column, including the point with coordinates $(r,r)$ where they intersect. However, it loses the old diagonal, that is to say the points aligned from the old upper left vertex $(1,r-1)$ to the old lower right vertex $(r-1,1)$. In fact, the region $T_{r}$ has a new diagonal, starting from $(1,r)$ and ending in $(r,1)$.
Now the equation tells us that, passing from $T_{r-1}$ to $T_r$:
we gain the weight of the point with coordinates $(r,r)$, as shown by the term $F(r,r)$;
we also gain the weights of the $r^{th}$ row and $r^{th}$ column, as shown by the summation $$\sum_{k=1}^{r-1} [f(k,r))+f(r,k)] $$
we lose the weight of the old diagonal, from point $(1,r-1)$ to $(r-1,1)$, as shown by the summation $$\sum_{k=1}^{r-1} f(k,r-k)$$