Question

Calculati: a. 1/3 x 9/5 x 15/7 x 21/9 x 27/11 x 33/13 : (3/1x4 + 3/4x7 +... +3/49x52) x 1 4/81
b. (1/2x3 + 1/3x4 +...+ 1/999x1000):(1/999-1/999x1000
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Answer 1

Răspuns:

33) a),b)

Explicație pas cu pas:

a)

$\frac{1}{3} \cdot \frac{9}{5} \cdot \frac{15}{7} \cdot \frac{21}{9} \cdot \frac{27}{11} \cdot \frac{33}{13} : \Big(\frac{3}{1\cdot 4} + \frac{3}{4 \cdot 7} + \frac{3}{7 \cdot 11} + ... + \frac{3}{46 \cdot 49} + \frac{3}{49 \cdot 52} \Big) \cdot 1 \frac{4}{81} =$

$= \frac{1}{\not3} \cdot \frac{3 \cdot \not 3}{\not5} \cdot \frac{3 \cdot\not 5}{\not7} \cdot \frac{3 \cdot\not 7}{\not9} \cdot \frac{3 \cdot\not 9}{\not11} \cdot \frac{3 \cdot\not 11}{13} : \Big(\frac{1}{1} -\not \frac{1}{4} +\not \frac{1}{4} -\not \frac{1}{7} +\not \frac{1}{7} -\not \frac{1}{11} + ... +\not \frac{1}{46} -\not \frac{1}{49} +\not \frac{1}{49} - \frac{1}{52} \Big) \cdot \frac{85}{81}$

$= \frac{15}{13} : \Big( \frac{1}{1} - \frac{1}{52} \Big) \cdot \frac{85}{81} = \frac{15}{13} : \frac{51}{52} \cdot \frac{85}{81} = \frac{15 \cdot 52 \cdot 85}{13 \cdot 51 \cdot 81} = \frac{\not3 \cdot 5 \cdot 4 \cdot\not 13 \cdot 5 \cdot\not 17}{\not13 \cdot\not 3 \cdot\not 17 \cdot 81} = \bf \frac{100}{81}$

b)

$\Big(\frac{1}{2\cdot 3} + \frac{1}{3 \cdot 4} + \frac{1}{4 \cdot 5} + ... + \frac{1}{998 \cdot 999} + \frac{1}{999 \cdot 1000} \Big) : \Big( \frac{1}{999} - \frac{1}{999 \cdot 1000} \Big) = \\ = \Big(\frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + \frac{1}{4} - \frac{1}{5} + ... + \frac{1}{998} - \frac{1}{999} + \frac{1}{999} - \frac{1}{1000} \Big) :  \frac{1000 - 1}{999 \cdot 1000} = \\ = \Big(\frac{1}{2} - \frac{1}{1000} \Big) : \frac{999}{999 \cdot 1000} =  \frac{500 - 1}{1000} : \frac{1}{1000} =  \frac{499}{1000} \cdot 1000 = \bf 499$