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Pe: 2013-10-29T07:55:40+02:00 In: In:

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Aflati numarul natural n care verifica egalitatiile:
a)2+2 la puterea 2 +2 la puterea 3 +….2 la puterea n=2 la puterea 55-2
b)2 la puterea n+2 la puterea n+1 +2 la puterea n +2=56.

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  1. Green eyes
    maestru (V)

    Salut,

    Punctul a:

    Notăm cu S suma din membrul stâng şi calculăm 2*S:

    \bl 2\cdot S=2^{\small 2}+2^{\small 3}+2^{\small 4}+...+2^{\small n}+2^{\small n+1}\\S=2+2^{\small 2}+2^{\small 3}+...+2^{\small n}

    Efectuăm scăderea 2*S-S membru cu membru:

    \bl S=2^{\small n+1}-2=2^{\small 55}-2,\;deci\;n+1=55\Rightarrow n=54.

    Punctul b:

    \bl 2^{\small n}+2^{\small n+1}+2^{\small n+2}=56,\;sau\;2^{\small n}+2\cdot 2^{\small n}+2^{\small 2}\cdot 2^{\small n}=56,\;sau\;2^{\small n}+2\cdot 2^{\small n}+4\cdot 2^{\small n}=56,\;sau\;(1+2+4)\cdot2^{\small n}=56,\\7\cdot2^{\small n}=56\Rightarrow 2^{\small n}=8,\;sau\;2^{\small n}=2^{\small 3}\Rightarrow n=3.

    Green eyes.

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