\(\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+....+\frac{99}{100!}<1\)chứng minh rằng
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Chứng minh rằng:a. frac{1}{3^2}+frac{2}{3^3}+frac{3}{3^4}+frac{4}{3^5}+...+frac{99}{3^{100}}+frac{100}{3^{101}} frac{1}{4}b.frac{1}{2}-frac{1}{4}+frac{1}{8}-frac{1}{16}+frac{1}{32}-frac{1}{64} frac{1}{3}c.frac{1}{3}-frac{2}{3^2}+frac{3}{3^3}-frac{4}{3^4}+...+frac{99}{3^{99}}-frac{100}{3^{100}} frac{1}{16}d. frac{1}{5^2}-frac{2}{5^3}+frac{3}{5^4}-frac{4}{5^5}+...+frac{99}{5^{100}}-frac{100}{5^{101}} frac{1}{36}
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Chứng minh rằng:
a. \(\frac{1}{3^2}+\frac{2}{3^3}+\frac{3}{3^4}+\frac{4}{3^5}+...+\frac{99}{3^{100}}+\frac{100}{3^{101}}< \frac{1}{4}\)
b.\(\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}+\frac{1}{32}-\frac{1}{64}< \frac{1}{3}\)
c.\(\frac{1}{3}-\frac{2}{3^2}+\frac{3}{3^3}-\frac{4}{3^4}+...+\frac{99}{3^{99}}-\frac{100}{3^{100}}< \frac{1}{16}\)
d. \(\frac{1}{5^2}-\frac{2}{5^3}+\frac{3}{5^4}-\frac{4}{5^5}+...+\frac{99}{5^{100}}-\frac{100}{5^{101}}< \frac{1}{36}\)
Tính nhanh :
A = \(\left(\frac{2}{3}+\frac{3}{4}+....+\frac{99}{100}\right)\cdot\left(\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+....+\frac{98}{99}\right)-\left(\frac{1}{2}+\frac{2}{3}+...+\frac{99}{100}\right)\cdot\left(\frac{2}{3}+\frac{3}{4}+...+\frac{98}{99}\right)\)
A=(2/3+3/4+...+99/100)x(1/2+2/3+3/4+...+98/99)-(1/2+2/3+...+99/100)x(2/3+3/4+4/5+...98/99)
ta cho nó dài hơn như sau
A=(2/3+3/4+4/5+5/6+....+98/99+99/100)
ta thấy các mẫu số và tử số giống nhau nên chệt tiêu các số
2:3:4:5...99 vậy ta còn các số 2/100
ta làm vậy với(1/2+2/3+3/4+.....+98/99) thi con 1/99
làm vậy với câu (1/2+2/3+...+99/100) thì ra la 1/100
vậy với (2/3+3/4+...+98/99) ra 2/99
xùy ra ta có 2/100.1/99-1/100.2/99=1/50x1/99-1/100x2/99=tự tinh nhe mình ngủ đây
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Chứng minh :
a) frac{1}{3}-frac{2}{3^2}+frac{3}{3^3}-frac{4}{3^4}+...+frac{99}{3^{99}}-frac{100}{3^{100}} frac{3}{16} frac{1}{3}-frac{2}{3^2}+frac{3}{3^3}+frac{4}{4^4}+...+frac{99}{3^{99}}-frac{100}{3^{100}} frac{3}{16}
b)frac{1}{41}+frac{1}{42}+frac{1}{43}+...+frac{1}{79}+frac{1}{80} frac{7}{12}
c) Cho Sfrac{3}{10}+frac{3}{11}+frac{3}{12}+frac{3}{13}+frac{3}{14}
Chứng minh 1 S 2
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Chứng minh :
a) \(\frac{1}{3}-\frac{2}{3^2}+\frac{3}{3^3}-\frac{4}{3^4}+...+\frac{99}{3^{99}}-\frac{100}{3^{100}}< \frac{3}{16}\) \(\frac{1}{3}-\frac{2}{3^2}+\frac{3}{3^3}+\frac{4}{4^4}+...+\frac{99}{3^{99}}-\frac{100}{3^{100}}< \frac{3}{16}\)
b)\(\frac{1}{41}+\frac{1}{42}+\frac{1}{43}+...+\frac{1}{79}+\frac{1}{80}< \frac{7}{12}\)
c) Cho \(S=\frac{3}{10}+\frac{3}{11}+\frac{3}{12}+\frac{3}{13}+\frac{3}{14}\)
Chứng minh \(1< S< 2\)
26 tháng 3 2016 lúc 16:34
Tính dãy số sau :
\(D=\frac{100-\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{99}+\frac{1}{100}\right)}{\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+...+\frac{98}{99}+\frac{99}{100}}\)
CMR \(\frac{1}{3}-\frac{2}{3^2}+\frac{2}{3^3}-\frac{4}{3^4}+......+\frac{99}{3^{99}}-\frac{100}{3^{100}}< \frac{3}{16}\)
\(A=\frac{1}{3}-\frac{2}{3^2}+\frac{3}{3^3}-\frac{4}{3^4}+.....+\frac{99}{3^{99}}-\frac{100}{3^{100}}\)
CMR: \(\frac{1}{3}-\frac{2}{3^2}=\frac{3}{3^3}-\frac{4}{4^4}+.......+\frac{99}{3^{99}}-\frac{100}{3^{100}}< \frac{3}{16}\)
Đặt A = \(\frac{1}{3}-\frac{2}{3^2}+\frac{3}{3^3}-\frac{4}{3^4}+....\frac{99}{3^{99}}-\frac{100}{3^{100}}\)
\(\Rightarrow3A=1-\frac{2}{3}+\frac{3}{3^2}-\frac{4}{3^3}+....\frac{99}{3^{98}}-\frac{100}{3^{99}}\)
\(\Rightarrow4A=A+3A=\left(1-\frac{1}{3}+\frac{1}{3^2}-\frac{1}{3^3}+.....\frac{1}{3^{98}}-\frac{1}{3^{99}}\right)\)
Đặt B = \(1-\frac{1}{3}+\frac{1}{3^2}-\frac{1}{3^3}+....\frac{1}{3^{98}}-\frac{1}{3^{99}}\Rightarrow4A< B\left(1\right)\)
\(\Rightarrow3B=3-1+\frac{1}{3}-\frac{1}{3^2}+....\frac{1}{3^{97}}-\frac{1}{3^{98}}\)
\(4B=B+3B=3-\frac{1}{3^{99}}< 3\Rightarrow4B< 3\Rightarrow B< \frac{3}{4}\left(2\right)\)
Từ (1) và (2) \(\Rightarrow4A< B< \frac{3}{4}\Rightarrow4A< \frac{3}{4}\Rightarrow A< \frac{3}{4}:4\Rightarrow A< \frac{3}{4}.\frac{1}{4}\Rightarrow A< \frac{3}{16}\)
=> đpcm.
\(B=\frac{1}{3}+\frac{2}{3^2}+\frac{3}{3^3}+\frac{4}{3^4}+...+\frac{99}{3^{99}}+\frac{100}{3^{100}}< \frac{3}{16}\)
Tìm kiếm trên câu tương tự thử nha, mấy dạng này khá nhiều =))
Tối rồi nên mik đi ngủ ko giải nx
Thông cảm
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\(\frac{1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{99}+\frac{1}{100}}{\frac{1}{99}+\frac{2}{98}+\frac{3}{97}+...+\frac{99}{1}+100}=?\)
\(\frac{1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{99}}{\frac{1}{99}+\frac{2}{98}+\frac{3}{97}+...+\frac{99}{1}}\)
\(=\frac{1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{99}}{\left(1+\frac{1}{99}\right)+\left(1+\frac{2}{98}\right)++...+\left(1+\frac{98}{2}\right)1}\)
\(=\frac{1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{99}}{\frac{100}{99}+\frac{100}{98}+...+\frac{100}{2}+\frac{100}{100}}\)
\(=\frac{1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{99}}{100\times\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{99}\right)}\)
\(=\frac{1}{100}\)
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CMR:
\(\frac{1}{3}-\frac{2}{3^2}+\frac{3}{3^3}-\frac{4}{3^4}+...+\frac{99}{3^{99}}-\frac{100}{3^{100}}
Đặt A=\(\frac{1}{3}\) - \(\frac{2}{3^2}\) +\(\frac{3}{3^3}\) - \(\frac{4}{3^4}\)+...+ \(\frac{99}{3^{99}}\) - \(\frac{100}{3^{100}}\)
=> 3A=1-\(\frac{2}{3}\) + \(\frac{3}{3^2}\) - \(\frac{4}{3^3}\)+...+\(\frac{99}{3^{98}}\) - \(\frac{100}{3^{99}}\)
=> 4A=1-\(\frac{1}{3}\)+\(\frac{1}{3^2}\)+...+\(\frac{1}{3^{98}}\) - \(\frac{1}{3^{99}}\)- \(\frac{100}{3^{100}}\)
=> 4A<1-\(\frac{1}{3}\)+\(\frac{1}{3^2}\)+...+\(\frac{1}{3^{98}}\) - \(\frac{1}{3^{99}}\) (1)
Đặt B=1-\(\frac{1}{3}\)+\(\frac{1}{3^2}\)+...+\(\frac{1}{3^{98}}\) - \(\frac{1}{3^{99}}\)
=> B=2+ \(\frac{1}{3}\) - \(\frac{1}{3^2}\) +...+\(\frac{1}{3^{97}}\) - \(\frac{1}{3^{98}}\)
=> 4B=B+3B=3-\(\frac{1}{3^{99}}\)<3 => A<\(\frac{3}{4}\) (2)
Từ (1) và (2) ta có: 4A<B<\(\frac{3}{4}\) => A<\(\frac{3}{16}\) => đpcm.
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