Tübitak Lise 1. Aşama 2010 Soru 30

[ERhan ERdoğan]

$N=\left \lfloor \dfrac{2}{5} \right \rfloor+\left \lfloor \dfrac{2^{2}}{5} \right \rfloor +\cdots+\left \lfloor \dfrac{2^{2009}}{5} \right \rfloor$ ise $2^{2010}$ un $N$ ile bölümünden kalan nedir?


$
\textbf{a)}\ 5034
\qquad\textbf{b)}\ 5032
\qquad\textbf{c)}\ 5031
\qquad\textbf{d)}\ 5028
\qquad\textbf{e)}\ 5024
$

[geo]

Yanıt: $\boxed{E}$

$$ \begin{array}{rcl}
N &=& \left \lfloor \dfrac{2^2}{10} \right \rfloor+\left \lfloor \dfrac{2^{3}}{10} \right \rfloor +\cdots+\left \lfloor \dfrac{2^{2010}}{10} \right \rfloor \\ \\
&=& \dfrac{2^2 - 4}{10} + \dfrac{2^3 - 8}{10} + \dfrac{2^4 - 6}{10} + \dfrac{2^5 - 2}{10} \\ \\
&& + \dfrac{2^6 - 4}{10} + \dfrac{2^7 - 8}{10} + \dfrac{2^8 - 6}{10} + \dfrac{2^9 - 2}{10} \\
&& + \cdots \\
&& + \dfrac{2^{2010} - 4}{10} \\ \\
&=& \dfrac { 2^2 + 2^3 + \cdots + 2^{2010} - 502\cdot ( 4 + 8 + 6 + 2) - 4} {10} \\ \\
\Longrightarrow 10N &=& 2^{2011} - 2^2 - 502 \cdot 20 - 4 \\
\Longrightarrow 5N&=& 2^{2010} - 502 \cdot 10 - 4 \\
\Longrightarrow 5N + 5024&=& 2^{2010}
\end{array}$$