2007 HCI C1 Lecture Test Q6 [Modified]

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2007 HCI C1 Lecture Test Q6 [Modified]

(i)

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]

(ii)

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

[3]

(iii)

Hence, evaluate

$\frac{1}{1\cdot 3}+\frac{1}{1\cdot 4}+\frac{1}{3\cdot 5}+\frac{1}{4\cdot 7}+\frac{1}{5\cdot 7}+\frac{1}{7\cdot 10}+…+\frac{1}{39\cdot 41}+\frac{1}{58\cdot 61}$.

[4]

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Published: 17th March 2023

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