2007 HCI C1 Lecture Test Q6 [Modified]

Verify that for any non-zero real constant, \(m\),

\(\frac{1}{m}\left( \frac{1}{mr+1-m}-\frac{1}{mr+1} \right)=\frac{1}{\left( mr+1-m \right)\left( mr+1 \right)}\).[1]

Hence, show that \(\sum\limits_{r=1}^{n}{\frac{1}{\left( mr+1-m \right)\left( mr+1 \right)}=\frac{n}{mn+1}}\).[3]

\(\frac{1}{1\cdot 3}+\frac{1}{1\cdot 4}+\frac{1}{3\cdot 5}+\frac{1}{4\cdot 7}+\frac{1}{5\cdot 7}+\frac{1}{7\cdot 10}+...+\frac{1}{39\cdot 41}+\frac{1}{58\cdot 61}\).[4]