Cho tổng : S= 2+2 mũ 2+2mũ3+...+2mũ2010. CHỨNG Minh rằng: Schia hết cho 3 ; S chia hết cho 5 ; S chia hết cho 7

Đáp án:

$\begin{array}{l}
S = 2 + {2^2} + {2^3} + ... + {2^{2010}}\\
 = \left( {2 + {2^2}} \right) + \left( {{2^3} + {2^4}} \right) + ... + \left( {{2^{2009}} + {2^{2010}}} \right)\\
 = 2.\left( {1 + 2} \right) + {2^3}.\left( {1 + 2} \right) + ... + {2^{2009}}.\left( {1 + 2} \right)\\
 = 2.3 + {2^3}.3 + ... + {2^{2009}}.3\\
 = \left( {2 + {2^3} + ... + {2^{2009}}} \right).3 \vdots 3\\
 \Leftrightarrow S \vdots 3\\
S = 2 + {2^2} + {2^3} + ... + {2^{2010}}\\
 = \left( {2 + {2^3}} \right) + \left( {{2^2} + {2^4}} \right) + ... + \left( {{2^{2008}} + {2^{2010}}} \right)\\
 = 2.\left( {1 + {2^2}} \right) + {2^2}.\left( {1 + {2^2}} \right) + ... + {2^{2008}}.\left( {1 + {2^2}} \right)\\
 = 2.5 + {2^2}.5 + ... + {2^{2008}}.5\\
 = \left( {2 + {2^2} + ... + {2^{2008}}} \right).5\\
 \Leftrightarrow S \vdots 5\\
S = 2 + {2^2} + {2^3} + ... + {2^{2010}}\\
 = \left( {2 + {2^2} + {2^3}} \right) + \left( {{2^4} + {2^5} + {2^6}} \right) + ... + \left( {{2^{2008}} + {2^{2009}} + {2^{2010}}} \right)\\
 = 2.\left( {1 + 2 + {2^2}} \right) + {2^4}.\left( {1 + 2 + {2^2}} \right) + ... + {2^{2008}}.\left( {1 + 2 + {2^2}} \right)\\
 = 2.7 + {2^4}.7 + ... + {2^{2008}}.7\\
 = \left( {2 + {2^4} + ... + {2^{2008}}} \right).7\\
 \Leftrightarrow S \vdots 7
\end{array}$