Find the sum of the series N=13+115+135.........100 terms

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Question

Find the sum of the series

N = \frac{1}{3} + \frac{1}{15} + \frac{1}{35} . . . . . . . . .100

terms

Answer: Option C
:
C
First term of the series

(\frac{1}{3}) = \frac{1}{(3 \times 1)} = \frac{1}{2} [1 - \frac{1}{3}]

Second term of the series

(\frac{1}{15}) = \frac{1}{(3 \times 5)} = \frac{1}{2} [\frac{1}{3} - \frac{1}{5}]

Third term of the series

(\frac{1}{35}) = \frac{1}{(7 \times 5)} = \frac{1}{2} [\frac{1}{5} - \frac{1}{7}]

..............................................................................
100th term of the series

(\frac{1}{199} \times 201) = \frac{1}{2} [\frac{1}{199} - \frac{1}{201}]

\frac{1}{2} [1 - \frac{1}{3} + \frac{1}{3} - \frac{1}{5} + \frac{1}{5} - \frac{1}{7} + \frac{1}{7} . . . . . . . . . . . . . . . . . . . + \frac{1}{199} - \frac{1}{201}]

\frac{1}{2} [1 - \frac{1}{201}] = \frac{200}{(2 \times 201)} = \frac{100}{201}

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