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»RADACINA PATRATA-gimnaziu

Data publicarii: 25 Ianuarie, 2012

EXEMPLUL 2

Suport teoretic:

Radacina patrata a unui numar natural, partea intreaga a unui numar real, suma de numere naturale.

Enunt:

Sa se calculeze suma:

S = $\sum_{k=1}^{k=2010}{[\sqrt{k}]},$ \sum_{k=1}^{k=2010}{[\sqrt{k}]},

unde [x] reprezinta partea intreaga a numarului real x.

Raspuns:

S = 59.114.

Rezolvare:

$\mathcal{S}=\Big([\sqrt{1}]+[\sqrt{2}]+[\sqrt{3}]\Big)+\Big([\sqrt{4}]+[\sqrt{5}]+[\sqrt{6}]+[\sqrt{7}]+[\sqrt{8}]\Big)+$ \mathcal{S}=\Big([\sqrt{1}]+[\sqrt{2}]+[\sqrt{3}]\Big)+\Big([\sqrt{4}]+[\sqrt{5}]+[\sqrt{6}]+[\sqrt{7}]+[\sqrt{8}]\Big)+

$+\Big([\sqrt{9}]+[\sqrt{10}]+[\sqrt{11}]+[\sqrt{12}]+[\sqrt{13}]+[\sqrt{14}]+[\sqrt{15}]\Big)+$ +\Big([\sqrt{9}]+[\sqrt{10}]+[\sqrt{11}]+[\sqrt{12}]+[\sqrt{13}]+[\sqrt{14}]+[\sqrt{15}]\Big)+

$+\Big([\sqrt{16}+\cdots+[\sqrt{24}]\Big)+\Big([\sqrt{25}]+\cdots+[\sqrt{35}]\Big)+\cdots+$ +\Big([\sqrt{16}+\cdots+[\sqrt{24}]\Big)+\Big([\sqrt{25}]+\cdots+[\sqrt{35}]\Big)+\cdots+

$+\Big([\sqrt{1849}]+\cdots+[\sqrt{1935}]\Big)+\Big([\sqrt{1936}]+\cdots+[\sqrt{2010}]\Big).$ +\Big([\sqrt{1849}]+\cdots+[\sqrt{1935}]\Big)+\Big([\sqrt{1936}]+\cdots+[\sqrt{2010}]\Big).