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MDCristi
Pe: 2013-05-15T17:22:10+03:00 In: In:

Binomul lui Newton

Determinati numarul de termeni rationali din dezvoltarile:
{(sqrt{2} + \sqrt[3]{4})}^{100}
Si as mai avea un exercitiu:

Suma coeficientilor primilor trei termeni ai dezvoltarii {(\sqrt{2^x} + \sqrt{2^{1-x}})}^{n}este 22
Daca T_3 + T_5 = 135
atunci aflati n si x

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  1. gunty
    maestru (V)

    1.

        \[ \begin{array}{l} T_{k + 1} = C_{100}^k \left( {\sqrt 2 } \right)^{100 - k} \cdot \left( {\sqrt[3]{4}} \right)^k = 2^{\frac{{100 - k}}{2}} \cdot 2^{\frac{{2k}}{3}} = 2^{\frac{{100 - k}}{2} + \frac{{2k}}{3}} = 2^{50 + \frac{k}{6}} \\ T_{k + 1} = 2^{50 + \frac{k}{6}} \in Q \Rightarrow k \vdots 6 \Rightarrow 17termeni.rationali. \\ \end{array} \]

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