First slide

Series of natural numbers

Question

Sum to 30 terms of the series $\frac{1}{1.2.3} + \frac{1}{2.3.4} + \frac{1}{3.4.5} + \dots$ is

Moderate

A

$\frac{116}{465}$
B

$\frac{495}{1984}$
C

$\frac{435}{791}$
D

$\frac{485}{791}$

Solution

$t_{r} = \frac{1}{r (r + 1) (r + 2)} = \frac{1}{2 r} - \frac{1}{r + 1} + \frac{1}{2 (r + 2)}$
[split into partial fractions]
$= \frac{1}{2} (\frac{1}{r} - \frac{1}{r + 1}) - \frac{1}{2} (\frac{1}{r + 1} - \frac{1}{r + 2})$
$\Rightarrow S_{n} = \sum_{r = 1}^{n} t_{r} = \frac{1}{2} (1 - \frac{1}{n + 1}) - \frac{1}{2} (\frac{1}{2} - \frac{1}{n + 2})$
$= \frac{1}{2} \frac{n}{n + 1} - \frac{n}{4 (n + 2)}$
Thus, $S_{30} = \frac{1}{2} \frac{30}{31} - \frac{30}{4 (32)} = \frac{30}{4} [\frac{64 - 31}{(31) (32)}] = \frac{495}{1984}$

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