Question

27. Find the sum
(1/2) + (1/6) + (1/12) + ...... + 1/156

Answer 1

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$\frac{1}{2} + \frac{1}{6} + \frac{1}{12} + ............ + \frac{1}{156}$

$= \frac{1}{1 \times 2} + \frac{1}{2 \times 3} + \frac{1}{3 \times 4} + ......... + \frac{1}{12 \times 13}$

$= \frac{(2 - 1)}{1 \times 2} + \frac{(3 - 2)}{2 \times 3} + \frac{(4 - 3)}{3 \times 4} + ......... + \frac{(13 - 12)}{12 \times 13}$

$= \frac{2}{ 1\times 2} - \frac{1}{1 \times 2} + \frac{3}{2 \times 3} - \frac{2}{2 \times 3} + \frac{4}{3 \times 4} - \frac{3}{3 \times 4} + ......... + \frac{13}{12 \times 13} - \frac{12}{12 \times 13}$

$= 1 - \frac{1}{ 2} + \frac{1}{2 } - \frac{1}{ 3} + \frac{1}{3 } - \frac{1}{4} + ......... + \frac{1}{12} - \frac{1}{13}$

$= 1 - \frac{1}{13}$

$= \frac{(13 - 1)}{13}$

$= \frac{12}{13}$

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