[问答题]
The following diagram shows the grap
j 7 ,yhn;mczc-03l xyu/fwq46lw 6 upch
a;90ge:icpep8mby 7j of $y=\arctan (2 x-3)+\frac{3 \pi}{4} $ for $x \in \mathbb{R}$ , with asymptotes at $y=\frac{\pi}{4}$ and $y=\frac{5 \pi}{4}$ .
1. Describe a sequence of transformations that transforms the graph of $y=\arctan x$ to the graph of $y=\arctan (2 x-3)+\frac{3 \pi}{4}$ for $x \in \mathbb{R}$ .
2. Show that $\arctan p-\arctan q \equiv \arctan \left(\frac{p-q}{1+p q}\right) $.
3. Verify that $\arctan (x+2)-\arctan (x+1)=\arctan \left(\frac{1}{(x+1)^{2}+(x+1)+1}\right)$ .
4. Using mathematical induction and the results from part (b) and (c), prove that
$\sum_{r=1}^{n} \arctan \left(\frac{1}{r^{2}+r+1}\right)=\arctan (n+1)-\frac{\pi}{4} \quad \text { for } n \in \mathbb{Z}^{+}$
