Matematică
ştrumfi28
2015-11-25 12:56:45
2/15+2/35+2/63+2/99+............+2/9999 Am nevoie unrgenta de acest ex⌚ Dau coroana
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Robeen
2015-11-25 14:08:42

[latex]dfrac{2}{15}+dfrac{2}{35}+dfrac{2}{63}+dfrac{2}{99}+...+dfrac{2}{9999}= \ \ =dfrac{2}{3cdot 5}+dfrac{2}{5cdot 7}+dfrac{2}{7cdot 9}+dfrac{2}{9cdot11}+...+dfrac{2}{99cdot 101} = \ \ =dfrac{5-3}{5cdot 3}+dfrac{7-5}{ 7cdot 5}+dfrac{9-7}{ 9cdot 7}+dfrac{11-9}{11cdot 9}+...+dfrac{101-99}{101cdot 99}= [/latex] [latex] =Big(dfrac{5}{5cdot 3}-dfrac{3}{5cdot 3}Big) + Big(dfrac{7}{7cdot 5}-dfrac{5}{7cdot 5}Big) +Big(dfrac{9}{9cdot 7}-dfrac{7}{9cdot 7}Big) +\ \ +Big(dfrac{11}{11cdot 9}-dfrac{9}{11cdot 9}Big) +...+Big(dfrac{101}{101cdot 99}- dfrac{99}{101cdot 99}Big)= \ \ \ = Big(dfrac{1}{3}-dfrac{1}{5}Big)+Big(dfrac{1}{5}-dfrac{1}{7}Big)+Big(dfrac{1}{7}-dfrac{1}{9}Big)+Big(dfrac{1}{9}-dfrac{1}{11}Big)+...+Big(dfrac{1}{99}-dfrac{1}{101}Big) =[/latex] [latex] = dfrac{1}{3}-dfrac{1}{5}+dfrac{1}{5}-dfrac{1}{7}+dfrac{1}{7}-dfrac{1}{9}+dfrac{1}{9}-dfrac{1}{11}+...+dfrac{1}{99}-dfrac{1}{101} = \ \ =dfrac{1}{3}-dfrac{1}{101} = dfrac{^{{ig{101}igslash}}1}{3}-dfrac{^{{ig{3}igslash}}1}{101} = dfrac{101}{101cdot 3}-dfrac{3}{101cdot 3} = dfrac{101-3}{101cdot 3} = dfrac{98}{303}[/latex]

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