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a)S=1+(-1/7)^1+(-1/7)^2+...+(-1/7)^2007
=>7S=7+(-1/7)^1+(1/7)^2+...+(-1/7)^2006
=>(7-1)S=6-(1/7)^2007
=>S=1-(-1/7^2007/6)
A = \(\frac{1}{2}\)\(-\)\(\frac{1}{2^2}\)\(+\)\(\frac{1}{2^3}\)\(-\)\(\frac{1}{2^4}\)\(+\)........... \(+\)\(\frac{1}{2^{99}}\)\(-\)\(\frac{1}{2^{100}}\)
2A = 1 - \(\frac{1}{2}\)+ \(\frac{1}{2^2}\)- \(\frac{1}{2^3}\)+.........+ \(\frac{1}{2^{98}}\)- \(\frac{1}{2^{99}}\)
2A + A =( 1 - \(\frac{1}{2}\)+ \(\frac{1}{2^2}\)- \(\frac{1}{2^3}\)+.........+ \(\frac{1}{2^{98}}\)- \(\frac{1}{2^{99}}\)) \(+\)( \(\frac{1}{2}\)\(-\)\(\frac{1}{2^2}\)\(+\)\(\frac{1}{2^3}\)\(-\)\(\frac{1}{2^4}\)\(+\)........... \(+\)\(\frac{1}{2^{99}}\)\(-\)\(\frac{1}{2^{100}}\))
3A = 1 \(-\) \(\frac{1}{2^{100}}\)
\(\Rightarrow\)A = \(\frac{1-\frac{1}{2^{100}}}{3}\)= \(\frac{1}{3}\)
Ta có : \(B=\frac{1}{2}-\frac{1}{2^2}+...-\frac{1}{2^{100}}\)
\(\Rightarrow2B=1-\frac{1}{2}+\frac{1}{2^2}-...-\frac{1}{2^{99}}\)
\(\Rightarrow2B+B=\left(1-\frac{1}{2}+\frac{1}{2^2}-...-\frac{1}{2^{99}}\right)+\left(\frac{1}{2}-\frac{1}{2^2}+...-\frac{1}{2^{100}}\right)\)
\(\Rightarrow3B=1-\frac{1}{2}+\frac{1}{2^2}-...-\frac{1}{2^{99}}+\frac{1}{2}-\frac{1}{2^2}+...-\frac{1}{2^{100}}\)
\(\Rightarrow3B=\left(1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{99}}\right)-\left(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{100}}\right)\)
\(\Rightarrow3B=1-\frac{1}{2^{100}}\)
\(\Rightarrow B=\frac{1-\frac{1}{2^{100}}}{3}\)
\(M=\frac{1}{2}-\frac{1}{2^2}+\frac{1}{2^3}-\frac{1}{2^4}+...+\frac{1}{2^{99}}-\frac{1}{2^{100}}\)
\(2M=1-\frac{1}{2}+\frac{1}{2^2}-\frac{1}{2^3}+...+\frac{1}{2^{98}}-\frac{1}{2^{99}}\)
\(2M+M=\left(1-\frac{1}{2}+\frac{1}{2^2}-\frac{1}{2^3}+...+\frac{1}{2^{98}}-\frac{1}{2^{99}}\right)+\left(\frac{1}{2}-\frac{1}{2^2}+\frac{1}{2^3}-\frac{1}{2^4}+...+\frac{1}{2^{99}}-\frac{1}{2^{100}}\right)\)
\(3M=1-\frac{1}{2^{100}}\)
\(M=\frac{1-\frac{1}{2^{100}}}{3}\)
Ta có 99/1+98/2+97/3+...+1/99=(98/2+1)+(97/3+1)+...+(1/99+1)+1
=100/2+100/3+...+100/99+100/100
=100(1/2+1/3=1/4+1/5+...+1/99+1/100)
Vậy (1/2+1/3+...+1/100)/((99/1+98/2+...+1/99)=1/100
ta có
S=1+ 1/2 +1/2^2 +..+1/2^100
=> S/2 -S=1/2+ 1/2^2+...+1/2^101-1-1/2-...1/2^100
=> -S/2=1/2^101-1
=> -S/2=(1-2^101)/2^101
=> S=-2*(1-2^101)/2^101
=> S=(2^101-1)/2^100