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parinte
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Pe: 2012-10-06T06:50:05+03:00 In: In:

algebra cls. a VII-a

nu le am cu demonstratiile, rog ajutor
1. daca a<b, demonstrati ca a/b < (a+1) / (b+1)
2. x=1/2X3/5X5/6X…..X119/120
y=2/3X4/5X6/7X…..X120/121
a) demonstrati ca x<y
b) demonstrati ca x<1/11
am facut xXy=1/121, dar demonstratiile a), b) ???
multumesc

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3 raspunsuri

  1. edy8
    maestru (V)

    parinte wrote:
    1. daca a<b, demonstrati ca a/b < (a+1) / (b+1)

    \it{\Large\bl a,b \in \mathbb{N}, a < b |_{\cdot ab} \Rightarrow a+ab < b+ab \Rightarrow a(b+1) < b(a+1) \Rightarrow \frac{a}{b} < \frac{a+1}{b+1}}

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  2. ali
    maestru (V)

    parinte wrote: a) demonstrati ca x<y


    \begin{array}{l} x = \frac{1}{2} \times \frac{3}{4} \times ... \times \frac{{119}}{{120}}\\ y = \frac{2}{3} \times \frac{4}{5} \times ... \times \frac{{120}}{{121}}\\ - - - - - - - - - - - - - - - - - \\ {\rm{Pronim de la inegalitatile evidente:}}\\ \left\{ \begin{array}{l} \frac{1}{2} < \frac{2}{3}\\ \frac{3}{4} < \frac{4}{5}\\ ..........\\ \frac{{119}}{{120}} < \frac{{120}}{{121}} \end{array} \right. \Rightarrow \frac{1}{2} \times \frac{3}{4} \times ... \times \frac{{119}}{{120}} < \frac{2}{3} \times \frac{4}{5} \times ... \times \frac{{120}}{{121}} \Leftrightarrow x < y \end{array}
    Pe baza acestei idei sunt formate probleme foarte interesante pentru clasa a 9 si a 11.

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  4. ali
    maestru (V)

    b) demonstrati ca x<1/11
    am facut xXy=1/121, dar demonstratiile a), b) ???


    Este destul … pe baza acestui rezultat+rezultatul de la sub-punctul a se dem. sub-punctul b:
    x < y\left| { \times x} \right. \Rightarrow {x^2} < xy = \frac{1}{{121}} \Rightarrow x < \sqrt {\frac{1}{{121}}} = \frac{1}{{11}}

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