Question

se considere numarul rațional a= 1/1.2+1/2.3+.....+1/9.10 aratati ca 0,8<a<1

Answer 1

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Answer 2

[tex]\it \dfrac{1}{k(k+1)} = \dfrac{1}{k} -\dfrac{1}{k+1} \\ \\ \\ a= \dfrac{1}{1\cdot2} + \dfrac{1}{2\cdot3} +\dfrac{1}{3\cdot4} +\ ...\ +\dfrac{1}{9\cdot10} = \dfrac{1}{1} -\dfrac{1}{2} +\dfrac{1}{2} -\dfrac{1}{3} + \\ \\ \\ + \dfrac{1}{3} -\dfrac{1}{4} +\ ...\ +\dfrac{1}{9} -\dfrac{1}{10} = 1-\dfrac{1}{10}= \dfrac{9}{10}=0,9[/tex]

$\it 0,8 \ \textless \ 0,9\ \textless \ 1 \Longrightarrow 0,8\ \textless \ a\ \textless \ 1$