CMR 1/3-2/3^2+3/2^3-....+100/2^100<3/16
cmr
100-(1+1/2+1/3+...+1/100)=1/2+2/3+3/4+....+99/100
\(=\left(1-1\right)+\left(1-\frac{1}{2}\right)+\left(1-\frac{1}{3}\right)+...+\left(1-\frac{1}{100}\right)\)
\(=\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+...+\frac{99}{100}\)
CMR 100 - ( 1 + 1/2 + 1/3 + ... + 1/100 ) = 1/2 + 2/3 + ... + 99/100.
CMR 1/3-2/3^2+3/3^3-.....+100/3^100
M=2^3-1/2^3+1.3^3-1/3^3+1. ... .100^3-1/100^3+1 CMR M>2/3
CMR
100-(1/2+1/3+1/4+...+1/100)=2/3+3/4+...+99/100
100=10*10
100=1000:10
100 câu nói hay về cuộc sống
cmr 100-(1+1/2+1/3+...+1/100)=1/2+2/3+...+99/100
Ta có :
\(100-\left(1+\frac{1}{2}+\frac{1}{3}+.....+\frac{1}{100}\right)=1.100-\left(1+\frac{1}{2}+\frac{1}{3}+....+\frac{1}{100}\right)\)
\(=\left(1-1\right)+\left(1-\frac{1}{2}\right)+\left(1-\frac{1}{3}\right)+.......+\left(1-\frac{1}{100}\right)\)
\(=\frac{1}{2}+\frac{2}{3}+.........+\frac{99}{100}\)
Vậy \(100-\left(1+\frac{1}{2}+\frac{1}{3}+......+\frac{1}{100}\right)=\frac{1}{2}+\frac{2}{3}+.....+\frac{99}{100}\left(ĐPCM\right)\)
cmr 1/2+2/2^2+3/2^3+...+100/2^100<2
Dat \(A=\frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3}+...+\frac{100}{2^{100}}\)
=> \(2A=1+1+\frac{3}{2^2}+...+\frac{100}{2^{99}}\)
=> \(2A-A=1+1+\frac{3}{2^2}+...+\frac{100}{2^{99}}-\left(\frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3}+...+\frac{100}{2^{100}}\right)\)
=> \(A=1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{99}}-\frac{100}{2^{100}}\)
Dat \(B=1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{99}}\)
=> \(2B=2+1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{98}}\)
=> \(2B-B=2+1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{98}}-\left(1+\frac{1}{2}+...+\frac{1}{2^{99}}\right)\)
=> \(B=2-\frac{1}{2^{99}}\)
=> \(A=2-\frac{1}{2^{99}}-\frac{100}{2^{100}}< 2\)
=> dpcm
CMR:(1+1/2+1/3+1/4+...+1/100)=1/2=2/3+3/4+...+99/100
cmr 1/3+2/3^2+•••+100/3^100 < 3/4