Question

1/1×2+1/2×3+1/3×4+...+1/99×100

3/1×4+3/4×7+3/7×10...+3/97×100


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Answer 1

Explicație pas cu pas:

$\frac{1}{1 \times 2} + \frac{1}{2 \times 3} + \frac{1}{3 \times 4} + ... + \frac{1}{99 \times 100} = ( \frac{1}{1} - \frac{1}{2}) + ( \frac{1}{2} - \frac{1}{3}) + ( \frac{1}{3} - \frac{1}{4}) + ... + ( \frac{1}{99} - \frac{1}{100}) = \frac{1}{1} - \frac{1}{2}+ \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + ... + \frac{1}{99} - \frac{1}{100} = \frac{1}{1} - \frac{1}{100} = \frac{99}{100}$

$\frac{3}{1 \times 4} + \frac{3}{4 \times 7} + \frac{3}{7 \times 10} + ... + \frac{3}{97 \times 100} = ( \frac{1}{1} - \frac{1}{4}) + (\frac{1}{4} - \frac{1}{7}) + (\frac{1}{7} - \frac{1}{10}) + ... + (\frac{1}{97} - \frac{1}{100}) = \frac{1}{1} - \frac{1}{4} + \frac{1}{4} - \frac{1}{7} + \frac{1}{7} - \frac{1}{10} + ... + \frac{1}{97} - \frac{1}{100} = \frac{1}{1} - \frac{1}{100} = \frac{99}{100}$