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15 tháng 3 2017

\(S=1+\frac{1}{2}\left(1+2\right)+\frac{1}{3}\left(1+2+3\right)+....+\frac{1}{100}\left(1+2+3+....+100\right)\)

\(=1+\frac{1}{2}.\frac{2.3}{2}+\frac{1}{3}.\frac{3.4}{2}+.....+\frac{1}{100}.\frac{100.101}{2}\)

\(=\frac{2}{2}+\frac{3}{2}+\frac{4}{2}+.....+\frac{101}{2}\)

\(=\frac{2+3+4+....+101}{2}\)

\(=\frac{\frac{101.102}{2}-1}{2}\)

\(=2575\)

Vậy \(S=2575\)

14 tháng 3 2017

\(1+\frac{1}{2}\left(1+2\right)+\frac{1}{3}\left(1+2+3\right)+.....+\frac{1}{100}\left(1+2+3+....+100\right)\)

\(=1+\frac{1}{2}.\frac{2\left(2+1\right)}{2}+\frac{1}{3}.\frac{3\left(3+1\right)}{2}+\frac{1}{4}.\frac{4\left(4+1\right)}{2}+.....+\frac{1}{100}.\frac{100\left(100+1\right)}{2}\)

\(=1+\frac{2+1}{2}+\frac{3+1}{2}+....+\frac{100+1}{2}\)

\(=\frac{2}{2}+\frac{3}{2}+\frac{4}{2}+....+\frac{101}{2}\)

\(=\frac{2+3+4+....+101}{2}\)

\(=\frac{\frac{101\left(101+1\right)}{2}-1}{2}=5150.5\)

19 tháng 8 2016

=> \(2S=1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{99}}\)

=>  \(2S-S=\left(1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{99}}\right)-\left(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{100}}\right)\)

=> S = \(1-\frac{1}{2^{100}}\)

19 tháng 8 2016

1/2.S =1/2 .(1/2+1/2^2+1/2^3 + ......+1/2^100)

1/2 . S=1/2^2 +1/2^3 +.....+1/2^101

1/2.S-S=1/2^2+1/2^3 +......+1/2^101 - (1/2 +1/2^2 +.....+1/2^1OO)

-1/2.S=1/2^101-1/2

S=(1/2^101-1/2):2

23 tháng 6 2016

a)S=2+22+23+...+2100

2S=2(2+22+23+...+2100)

2S=22+23+...+2101

2S-S=(22+23+...+2101)-(2+22+23+...+2100)

S=2101-2

b)\(P=\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{100}}\)

\(3P=3\left(\frac{1}{3}+\frac{1}{3^2}+....+\frac{1}{3^{100}}\right)\)

\(3P=1+\frac{1}{3}+...+\frac{1}{3^{99}}\)

\(3P-P=\left(1+\frac{1}{3}+...+\frac{1}{3^{99}}\right)-\left(\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{99}}\right)\)

\(2P=1-\frac{1}{3^{100}}\)

\(P=\left(1-\frac{1}{3^{100}}\right):2\)

23 tháng 6 2016

cái này lớp 6 mà bạn :D