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Find sum of \(\rm \frac{1}{{1 \times 2}} + \frac{1}{{2 \times 3}} + \frac{1}{{3 \times 4}} +...+\frac{1}{{n \times (n+1)}} \)1. n(n + 1)2. \(\rm \frac {n}{n+1}\)3. \(\rm \frac {2n}{n+1}\)4. None of these |
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Answer» Correct Answer - Option 2 : \(\rm \frac {n}{n+1}\) Calculation: \(\rm \frac{1}{{1 \times 2}} + \frac{1}{{2 \times 3}} + \frac{1}{{3 \times 4}} +...+\frac{1}{{n \times (n+1)}} \) \(\rm = \frac{{2\; - \;1}}{{1 \times 2}} + \frac{{3\; - \;2}}{{2 \times 3}} + \frac{{4\; - \;3}}{{3 \times 4}} +... + \frac{{(n+1)\; - \;n}}{{n \times (n+1)}}\) \(\rm = \frac{1}{1} - \frac{1}{2} + \;\frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} +... + \frac{1}{n} - \frac{1}{n+1}\) \(\rm = 1 - \frac {1}{n+1}\) \(\rm = \frac {n+1 -1}{n+1}\) \(\rm = \frac {n}{n+1}\) |
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