a,A = 1 + 2 + 22 + 23 +.... + 22013 + 22014
2A = 2 + 22 + 23 + ...... + 22013 + 22014 + 22015
A = ( 2 + 22 + 23 + ..... + 22013 + 22014 + 22015 ) - ( 1 + 2 + 22 + 23 + ..... + 22013 + 22014 )
A = 22015 - 1
b, A = 1 + 2 + 22 + 23 + ... + 22013 + 22014
= ( 1 + 2 + 22 + 23 + 24 ) + .... + ( 22010 + 22011 + 22012 + 22013 + 22014 )
= 31 + ..... + 22010.( 1 + 2 + 22 + 23 + 24 )
= 31 + ..... + 22010 . 31
= 31.1 + ..... + 22010 . 31
= 31. ( 1 + .... + 22010 ) chia hết cho 31
=> A chia hết cho 31
a) \(A=1+2+2^2+2^3+....+2^{2014}\)
\(\Leftrightarrow\)\(2A=2+2^2+2^3+2^4+...+2^{2015}\)
\(\Leftrightarrow\)\(2A-A=\left(2+2^2+2^3+...+2^{2015}\right)-\left(1+2+2^2+...+2^{2014}\right)\)
\(\Leftrightarrow\)\(A=2^{2015}-1\)
b) \(A=1+2+2^2+2^3+...+2^{2014}\)
\(=\left(1+2+2^2+2^3+2^4\right)+\left(2^5+2^6+2^7+2^8+2^9\right)\)\(+...+\left(2^{2010}+2^{2011}+2^{2012}+2^{2013}+2^{2014}\right)\)
\(=\left(1+2+2^2+2^3+2^4\right)+2^5\left(1+2+2^2+2^3+2^4\right)\)\(+...+2^{2010}\left(1+2+2^2+2^3+2^4\right)\)
\(=\left(1+2+2^2+2^3+2^4\right)\left(1+2^5+...+2^{2010}\right)\)
\(=31\left(1+2^5+...+2^{2010}\right)\) \(⋮31\)
a; A = 1 + 2+ 22 + 23 +..................+ 22013 + 22014
2A = 2+ 22 + 23 +..................+ 22013 + 22015
2A - A = [ 2+ 22 + 23 +..................+ 22013 + 22015 ] - [ 1 + 2+ 22 + 23 +..................+ 22013 + 22014 ]
A = 22015 - 1
b; A= 1 + 2+ 22 + 23 +..................+ 22013 + 22014
A = [ 1 + 2+ 22 + 23 ] + [ 24 + 25 + 26 + 27 ] +[ 28+29+210+211 ]+..................+ [ 22011+ 22012+22013+ 22014 ]
A = 31 + 23 [1 + 2 +22 + 23 + 24 ] + 28 [ 1 + 2+ 22 + 23 ] + ................+ 22011 [ 1 + 2+ 22 + 23 ]
A = 31 + 23 .31 + 28 . 31 +....................+ 22011 . 31
A = 31 [ 23 + 28 +..........+ 22011 ]
Mà 31 chia hết cho 31 => 31 [ 23 + 28 +..........+ 22011 ] chia hết cho 31 hay A chia hết cho 31
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