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Simplify (1 - 1/4)(1 - 1/9)(1 - 1/16) ... (1 - 1/10000).

 May 19, 2020
 #1
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Simplify

 

\(\left(1 - \dfrac{1}{4} \right) \left(1 - \dfrac{1}{9} \right) \left(1 - \dfrac{1}{16} \right) \left(1 - \dfrac{1}{25} \right) \left(1 - \dfrac{1}{36} \right) \times \cdots \times \left(1 - \dfrac{1}{9801}\right) \left(1 - \dfrac{1}{10000}\right)\).

 

\(\small{ \begin{array}{|rcll|} \hline && \mathbf{\left(1 - \dfrac{1}{4} \right) \left(1 - \dfrac{1}{9} \right) \left(1 - \dfrac{1}{16} \right) \left(1 - \dfrac{1}{25} \right) \left(1 - \dfrac{1}{36} \right)\cdots \left(1 - \dfrac{1}{9801}\right) \left(1 - \dfrac{1}{10000}\right)} \\ \\ &=& \left(\dfrac{4-1}{4} \right) \left(\dfrac{9-1}{9} \right) \left(\dfrac{16-1}{16} \right) \left(\dfrac{25-1}{25} \right) \left(\dfrac{36-1}{36} \right)\cdots \left(\dfrac{9801-1}{9801}\right) \left(\dfrac{10000-1}{10000}\right) \\ \\ &=& \dfrac{(2-1)(2+1)}{2*2} \cdot \dfrac{(3-1)(3+1)}{3*3} \cdot \dfrac{(4-1)(4+1)}{4*4} \cdot \dfrac{(5-1)(5+1)}{5*5} \cdot \dfrac{(6-1)(6+1)}{6*6}\cdots \dfrac{(99-1)(99+1)}{99*99}\cdot \dfrac{(100-1)(100+1)}{100*100} \\ \\ &=& \dfrac{ 1*3 }{2*2} \cdot \dfrac{ 2*4 }{3*3} \cdot \dfrac{ 3*5 }{4*4} \cdot \dfrac{ 4*6 }{5*5} \cdot \dfrac{ 5*7 }{6*6}\cdots \dfrac{ 98*100 }{99*99}\cdot \dfrac{ 99*101 }{100*100} \\ \\ &=& \dfrac{ 1*\color{red}3 }{2*2} \cdot \dfrac{ {\color{green}2}*4 }{3*3} \cdot \dfrac{ 3*5 }{4*4} \cdot \dfrac{ 4*6 }{5*5} \cdot \dfrac{ 5*7 }{6*6}\cdots \dfrac{ 98*100 }{99*99}\cdot \dfrac{ 99*101 }{100*100} \quad | \quad \text{change red and green} \\ \\ &=& \dfrac{ 1*\color{green}2 }{2*2} \cdot \dfrac{ {\color{red}3}*4 }{3*3} \cdot \dfrac{ 3*5 }{4*4} \cdot \dfrac{ 4*6 }{5*5} \cdot \dfrac{ 5*7 }{6*6}\cdots \dfrac{ 98*100 }{99*99}\cdot \dfrac{ 99*101 }{100*100} \\ \\ &=& \dfrac{ 1*2}{2*2} \cdot \dfrac{ 3*{\color{red}4} }{3*3} \cdot \dfrac{ {\color{green}3} *5 }{4*4} \cdot \dfrac{ 4*6 }{5*5} \cdot \dfrac{ 5*7 }{6*6}\cdots \dfrac{ 98*100 }{99*99}\cdot \dfrac{ 99*101 }{100*100} \quad | \quad \text{change red and green} \\ \\ &=& \dfrac{ 1*2}{2*2} \cdot \dfrac{ 3*{\color{green}3} }{3*3} \cdot \dfrac{ {\color{red}4} *5 }{4*4} \cdot \dfrac{ 4*6 }{5*5} \cdot \dfrac{ 5*7 }{6*6}\cdots \dfrac{ 98*100 }{99*99}\cdot \dfrac{ 99*101 }{100*100} \\ \\ &=& \dfrac{ 1*2 }{2*2} \cdot \dfrac{ 3*3 }{3*3} \cdot \dfrac{ 4*{\color{red}5} }{4*4} \cdot \dfrac{ {\color{green}4} *6 }{5*5} \cdot \dfrac{ 5*7 }{6*6}\cdots \dfrac{ 98*100 }{99*99}\cdot \dfrac{ 99*101 }{100*100} \quad | \quad \text{change red and green} \\ \\ &=& \dfrac{ 1*2 }{2*2} \cdot \dfrac{ 3*3 }{3*3} \cdot \dfrac{ 4*{\color{green}4} }{4*4} \cdot \dfrac{ {\color{red}5} *6 }{5*5} \cdot \dfrac{ 5*7 }{6*6}\cdots \dfrac{ 98*100 }{99*99}\cdot \dfrac{ 99*101 }{100*100} \\ \\ &=& \dfrac{ 1*2 }{2*2} \cdot \dfrac{ 3*3 }{3*3} \cdot \dfrac{ 4*4 }{4*4} \cdot \dfrac{ 5*{\color{red}6} }{5*5} \cdot \dfrac{ {\color{green}5}*7 }{6*6}\cdots \dfrac{ 98*100 }{99*99}\cdot \dfrac{ 99*101 }{100*100} \quad | \quad \text{change red and green} \\ \\ &=& \dfrac{ 1*2 }{2*2} \cdot \dfrac{ 3*3 }{3*3} \cdot \dfrac{ 4*4 }{4*4} \cdot \dfrac{ 5*{\color{green}5} }{5*5} \cdot \dfrac{ {\color{red}6}*7 }{6*6}\cdots \dfrac{ 98*100 }{99*99}\cdot \dfrac{ 99*101 }{100*100} \\ \\ &=& \dfrac{ 1*2 }{2*2} \cdot \dfrac{ 3*3 }{3*3} \cdot \dfrac{ 4*4 }{4*4} \cdot \dfrac{ 5*5 }{5*5} \cdot \dfrac{ 6*7 }{6*6}\cdots \dfrac{ 98*100 }{99*99}\cdot \dfrac{ 99*101 }{100*100} \\ \\ && \ldots \\ \\ &=& \dfrac{ 1*2 }{2*2} \cdot \dfrac{ 3*3 }{3*3} \cdot \dfrac{ 4*4 }{4*4} \cdot \dfrac{ 5*5 }{5*5} \cdot \dfrac{ 6*6 }{6*6}\cdots \dfrac{ 99*{\color{red}100} }{99*99}\cdot \dfrac{ {\color{green}99}*101 }{100*100} \quad | \quad \text{change red and green} \\ \\ && \ldots \\ &=& \dfrac{ 1*2 }{2*2} \cdot \dfrac{ 3*3 }{3*3} \cdot \dfrac{ 4*4 }{4*4} \cdot \dfrac{ 5*5 }{5*5} \cdot \dfrac{ 6*6 }{6*6}\cdots \dfrac{ 99*{\color{green}99} }{99*99}\cdot \dfrac{ {\color{red}100}*101 }{100*100} \\ \\ &=& \dfrac{ 1*2 }{2*2} \cdot \dfrac{ 3*3 }{3*3} \cdot \dfrac{ 4*4 }{4*4} \cdot \dfrac{ 5*5 }{5*5} \cdot \dfrac{ 6*6 }{6*6}\cdots \dfrac{ 99*99 }{99*99}\cdot \dfrac{ 100*101 }{100*100} \\ \\ &=& \dfrac{1}{2}\cdot \dfrac{101}{100} \\\\ &=& \mathbf{\dfrac{101}{200}} \\ \hline \end{array} }\)

 

laugh

 May 19, 2020
 #2
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3/4 * 8/9 * 15/16 * 24/25 * 35/36 * 48/49 *......* 9999/10000

 

product_(n=1)^99 (n (n + 2))/(n + 1)^2 = 101/200

 May 19, 2020

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