1
A=1+2+2^2+2^3+2^4+…….+2^56
Tính a chứng minh A chia hết cho 7
B=-2+4+(-6)+8+(-10)+…+100
Tính B
C=1+3^2+3^4+3^6+…+3^100
Chứng minh C chia hết cho 91
1 A=1+2+2^2+2^3+2^4+…….+2^56 Tính a chứng minh A chia hết cho 7 B=-2+4+(-6)+8+(-10)+…+100 Tính B C=1+3^2+3^4+3^6+…+3^100 Chứng minh C chia h
By Valerie
Bạn tham khảo :
$A$ $= 1 +2+2^2+2^3+2^4+…….+2^{56} = ( 1 + 2 + 2^2 ) + ( 2^2 + 2^3 + 2^4) + …. + (2^{54} + 2^{55} + 2^{56} ) = ( 1+2+2^2) . 1 + 2^2( 1 + 2 + 2^2) …. + 2^{54} ( 1 + 2 + 2^2) = 7 + 7. 2^2 + … + 7. 1^54 = 7 ( 1 + 2^2 + … + 2^{54})$
Vì $7 \vdots 7 ⇒ 7 ( 1 + 2^2 + … + 2^54 ) \vdots 7 ⇒A = 1 +2+2^2+2^3+2^4+…….+2^56 \vdots 7$
$B = – 2+4+(-6)+8+(-10)+…+100 = (-2 + 4 ) + ( -6 + 8 ) + (-10 + 12 ) + … + (-98 + 100 ) = 2 + 2 + 2 + … + 2 = 2.25 = 50 $
$C =1+3^2+3^4+3^6+…+3^{100} = ( 1+ 3^2 + 3^4 ) + (3^6 + 3^8 + 3^10 ) + …+ (3^{96} +3^{98} + 3^{100} = 91 . 3^6 ( 1 + 2 + 3^4) + … + 3^{96} ( 1 + 2 + 3^4) = 91 . 3^6 . 91 + … 3^{96} . 91 = 91 ( 1 + 3^6 + … + 3^{96} )$
Vì $91 \vdots 91 ⇒ 91 ( 1 + 3^6 + … + 3^96 ) \vdots 91 ⇒C =1+3^2+3^4+3^6+…+3^{100} \vdots 91$
Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
A = 1 + 2 + {2^2} + {2^3} + {2^4} + …. + {2^{56}}\\
= \left( {1 + 2 + {2^2}} \right) + \left( {{2^3} + {2^4} + {2^5}} \right) + …. + \left( {{2^{54}} + {2^{55}} + {2^{56}}} \right)\\
= \left( {1 + 2 + {2^2}} \right) + {2^3}\left( {1 + 2 + {2^2}} \right) + …. + {2^{54}}\left( {1 + 2 + {2^2}} \right)\\
= \left( {1 + 2 + {2^2}} \right)\left( {1 + {2^3} + {2^6} + …. + {2^{54}}} \right)\\
= 7.\left( {1 + {2^3} + {2^6} + … + {2^{54}}} \right) \vdots 7
\end{array}\)
\(\begin{array}{l}
B = \left( { – 2} \right) + 4 + \left( { – 6} \right) + 8 + \left( { – 10} \right) + …. + 100\\
= \left( { – 2 + 4} \right) + \left( { – 6 + 8} \right) + \left( { – 10 + 12} \right) + ….. + \left( { – 98 + 100} \right)\\
= 2 + 2 + 2 + …. + 2\\
= 2.25\\
= 50\\
C = 1 + {3^2} + {3^4} + {3^6} + …. + {3^{100}}\\
= \left( {1 + {3^2} + {3^4}} \right) + \left( {{3^6} + {3^8} + {3^{10}}} \right) + ….. + \left( {{3^{96}} + {3^{98}} + {3^{100}}} \right)\\
= \left( {1 + {3^2} + {3^4}} \right) + {3^6}\left( {1 + {3^2} + {3^4}} \right) + …. + {3^{96}}\left( {1 + {3^2} + {3^4}} \right)\\
= \left( {1 + {3^2} + {3^4}} \right)\left( {1 + {3^6} + {3^{12}} + …. + {3^{96}}} \right)\\
= 91\left( {1 + {3^6} + {3^{12}} + …. + {3^{96}}} \right) \vdots 91
\end{array}\)