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DT
2
DC
3 tháng 7 2020
1 ... 1/1 x 1 + 1/2 x 2 + 1/3 x 3 + ... + 1/100 x 100
1 ... 1+1/2x2+1/3x3+...+1/100x100
1=1/1x1+1/2x2+1/3x3+...+1/100x100
EC
1
NT
0
N
1
NL
Nguyễn Lê Phước Thịnh
CTVHS
8 tháng 1
Ta có: \(\frac{1}{3\times3}<\frac{1}{2\times3}=\frac12-\frac13\)
\(\frac{1}{4\times4}<\frac{1}{3\times4}=\frac13-\frac14\)
...
\(\frac{1}{100\times100}<\frac{1}{99\times100}=\frac{1}{99}-\frac{1}{100}\)
Do đó: \(\frac{1}{3\times3}+\frac{1}{4\times4}+\cdots+\frac{1}{100\times100}<\frac12-\frac13+\frac13-\frac14+\cdots+\frac{1}{99}-\frac{1}{100}<\frac12\)
=>\(\frac{1}{2\times2}+\frac{1}{3\times3}+\cdots+\frac{1}{100\times100}<\frac{1}{2\times2}+\frac12\)
=>\(P<\frac14+\frac12=\frac34\)
NM
1
10 tháng 8 2015
Ta có:
\(\frac{1}{2x2}<\frac{1}{1.2}\)
\(\frac{1}{3x3}<\frac{1}{2.3}\)
\(...\)
\(\frac{1}{2015x2015}<\frac{1}{2014x2015}\)
\(\Rightarrow\frac{1}{2x2}+\frac{1}{3x3}+...+\frac{1}{2015x2015}<\frac{1}{1x2}+\frac{1}{2x3}+...+\frac{1}{2014x2015}\)
\(\Rightarrow\frac{1}{2x2}+\frac{1}{3x3}+...+\frac{1}{2015x2015}<1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{2014}-\frac{1}{2015}\)
\(\Rightarrow\frac{1}{2x2}+\frac{1}{3x3}+...+\frac{1}{2015x2015}<1-\frac{1}{2015}<1\)
\(\Rightarrow\)Đpcm
NM
2
XO
4 tháng 8 2020
\(E=\frac{1}{2.2}+\frac{1}{3.3}+\frac{1}{4.4}+...+\frac{1}{49.49}\)
Ta có \(\frac{1}{2.2}>\frac{1}{2.3}\)
\(\frac{1}{3.3}>\frac{1}{3.4}\)
...
\(\frac{1}{49.49}>\frac{1}{49.50}\)
=> \(E=\frac{1}{2.2}+\frac{1}{3.3}+...+\frac{1}{49.49}>\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{49.50}\)
\(=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{49}-\frac{1}{50}=\frac{1}{2}-\frac{1}{50}=\frac{24}{50}=\frac{12}{25}=F\)
=> E > F
PV
0
2870 ĐÚNG NHÉ BẠN !
\(C=1x\left(2-1\right)+2x\left(3-1\right)+3x\left(4-1\right)+...+20x\left(21-1\right)\)
\(=1x2-1+2x3-2+3x4-3+...+20x21-20\)
\(=\left(1x2+2x3+3x4+...+20x21\right)-\left(1+2+3+...+20\right)\)
\(A=1x2+2x3+3x4+...+21x21\)
\(3xA=1x2x3+2x3x3+3x4x3+...+20x21x3\)
\(3xA=1x2x3+2x3x\left(4-1\right)+3x4x\left(5-2\right)+...+20x21x\left(22-19\right)\)
\(3xA=1x2x3-1x2x3+2x3x4-2x3x4+3x4x5-...-19x20x21+20x21x22\)
\(3xA=20x21x22\Rightarrow A=20x7x22=3080\)
\(B=1+2+3+...+20=\frac{20x\left(1+20\right)}{2}=210\)
\(C=A-B=3080-210=2870\)
Bài làm:
\(1+2.2+3.3+...+20.20\)
\(=1.\left(2-1\right)+2\left(3-1\right)+3\left(4-1\right)+...+20\left(21-1\right)\)
\(=1.2-1+2.3-2+3.4-3+...+20.21-20\)
\(=\left(1.2+2.3+3.4+...+20.21\right)-\left(1+2+3+...+20\right)\)
\(=\frac{1}{3}\left(1.2.3+2.3.3+...+20.21.3\right)-\frac{\left(20+1\right).20}{2}\)
\(=\frac{1}{3}\left[1.2.3+2.3.\left(4-1\right)+...+20.21.\left(22-19\right)\right]-210\)
\(=\frac{1}{3}\left(1.2.3-1.2.3+2.3.4-...-19.20.21+20.21.22\right)-210\)
\(=\frac{1}{3}.20.21.22-210\)
\(=3080-210\)
\(=2870\)
Học tốt!!!!