مقدار A+B

rezashiri · 1389/7/21

اگر داشته باشیم :

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$B=\frac{3}{28}+\frac{3}{70}+\frac{3}{130}+...+\frac{3}{1978}$

$A=\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+...+\frac{1}{240}$

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مقدار

$A+B$

را بیابید.

Sepehrmousavi · 1389/7/22

$A = \frac{1}{2} + \frac{1}{6} + \frac{1}{12} + ... + \frac{1}{240}$

$A = \frac{1}{1 \times 2} + \frac{1}{2 \times 3} + \frac{1}{3 \times 4} + ... + \frac{1}{15 \times 16}$

$A = (\frac{1}{1} - \frac{1}{2}) + (\frac{1}{2} - \frac{1}{3}) + (\frac{1}{3} - \frac{1}{4}) + ... + (\frac{1}{15} - \frac{1}{16})$

$A = (\frac{1}{1} - \frac{1}{16}) = \frac{15}{16}$

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$B = \frac{3}{28} + \frac{3}{70} + \frac{3}{130} + ... + \frac{3}{1978}$

$B = \frac{3}{4 \times 7} + \frac{3}{ 7 \times 10} + \frac{3}{10 \times 13} + ... + \frac{3}{43 \times 46}$

$B = (\frac{1}{4} - \frac{1}{7}) + (\frac{1}{7} - \frac{1}{10}) + (\frac{1}{10} - \frac{1}{13}) + ... + (\frac{1}{43} - \frac{1}{46})$

$B = (\frac{1}{4} - \frac{1}{46}) = \frac{42}{184} = \frac{21}{92}$

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$A + B = (\frac{15}{16} + \frac{21}{92}) = \frac{1380+336}{1472} = \frac{1716}{1472} = \frac{429}{368}$

AidinT · 1389/7/22

اصولاً روش حل کردن این مسأله ها ، یافتن اعدادیه که ضربشون مخرج و اختلافشون صورت کسره!