B= 3+3^3+3^5+….+ 3^1991.Chứng minh B chia hết 13 và 41 02/09/2021 Bởi Piper B= 3+3^3+3^5+….+ 3^1991.Chứng minh B chia hết 13 và 41
Tổng trên có (1991-1):2+1 = 996 số hạng Ta có: \(\begin{array}{l}B = 3 + {3^3} + {3^5} + … + {3^{1991}}\\B = \left( {3 + {3^3} + {3^5}} \right) + \left( {{3^6} + {3^7} + {3^8}} \right) + … + \left( {{3^{1989}} + {3^{1990}} + {3^{1991}}} \right)\\B = 3\left( {1 + {3^2} + {3^4}} \right) + {3^6}\left( {1 + {3^2} + {3^4}} \right) + … + {3^{1989}}\left( {1 + {3^2} + {3^4}} \right)\\B = \left( {1 + {3^2} + {3^4}} \right)\left( {3 + {3^6} + … + {3^{1989}}} \right)\\B = 91\left( {3 + {3^6} + … + {3^{1989}}} \right)\end{array}\) Vì: \(91\,\, \vdots \,\,13 \Rightarrow B\,\, \vdots \,\,13\) Ta có: \(\begin{array}{l}B = 3 + {3^3} + {3^5} + … + {3^{1991}}\\B = \left( {3 + {3^3} + {3^5} + {3^6}} \right) + \left( {{3^7} + {3^8} + {3^9} + {3^{10}}} \right) + … + \left( {{3^{1988}} + {3^{1989}} + {3^{1990}} + {3^{1991}}} \right)\\B = 3\left( {1 + {3^2} + {3^4} + {3^6}} \right) + {3^7}\left( {1 + {3^2} + {3^4} + {3^6}} \right) + … + {3^{1988}}\left( {1 + {3^2} + {3^4} + {3^6}} \right)\\B = \left( {1 + {3^2} + {3^4} + {3^6}} \right)\left( {3 + {3^7} + … + {3^{1988}}} \right)\\B = 823\left( {3 + {3^7} + … + {3^{1988}}} \right)\end{array}\) Vì \(820\,\, \vdots \,\,41 \Rightarrow B\,\, \vdots \,\,41\) Bình luận
Tổng trên có (1991-1):2+1 = 996 số hạng
Ta có:
\(\begin{array}{l}
B = 3 + {3^3} + {3^5} + … + {3^{1991}}\\
B = \left( {3 + {3^3} + {3^5}} \right) + \left( {{3^6} + {3^7} + {3^8}} \right) + … + \left( {{3^{1989}} + {3^{1990}} + {3^{1991}}} \right)\\
B = 3\left( {1 + {3^2} + {3^4}} \right) + {3^6}\left( {1 + {3^2} + {3^4}} \right) + … + {3^{1989}}\left( {1 + {3^2} + {3^4}} \right)\\
B = \left( {1 + {3^2} + {3^4}} \right)\left( {3 + {3^6} + … + {3^{1989}}} \right)\\
B = 91\left( {3 + {3^6} + … + {3^{1989}}} \right)
\end{array}\)
Vì: \(91\,\, \vdots \,\,13 \Rightarrow B\,\, \vdots \,\,13\)
Ta có:
\(\begin{array}{l}
B = 3 + {3^3} + {3^5} + … + {3^{1991}}\\
B = \left( {3 + {3^3} + {3^5} + {3^6}} \right) + \left( {{3^7} + {3^8} + {3^9} + {3^{10}}} \right) + … + \left( {{3^{1988}} + {3^{1989}} + {3^{1990}} + {3^{1991}}} \right)\\
B = 3\left( {1 + {3^2} + {3^4} + {3^6}} \right) + {3^7}\left( {1 + {3^2} + {3^4} + {3^6}} \right) + … + {3^{1988}}\left( {1 + {3^2} + {3^4} + {3^6}} \right)\\
B = \left( {1 + {3^2} + {3^4} + {3^6}} \right)\left( {3 + {3^7} + … + {3^{1988}}} \right)\\
B = 823\left( {3 + {3^7} + … + {3^{1988}}} \right)
\end{array}\)
Vì \(820\,\, \vdots \,\,41 \Rightarrow B\,\, \vdots \,\,41\)