(1/2 +1/3 + 1/4 +....+1/80) x> 1/79 + 2/78 + 3/77 +...+ 78/2 + 79/1)
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![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(1+\left(-2\right)+3+\left(-4\right)+5+...+77+\left(-78\right)+79+\left(-80\right)\)
\(=\left[1+\left(-2\right)\right]+\left[3+\left(-4\right)\right]+\left[5+\left(-6\right)\right]+...+\left[77+\left(-78\right)\right]+\left[79+\left(-80\right)\right]\)
\(=-1+\left(-1\right)+\left(-1\right)+...+\left(-1\right)\)
Từ 1 -> 80 có số số hạng là: (80-1):1+1=80 số hạng
=> Có 80:2=40 số -1
\(=-1\cdot40=-40\)
Đặt: \(A=1+\left(-2\right)+3+\left(-4\right)+...+77+\left(-78\right)+79+\left(-80\right)\)
\(\Rightarrow A=1-2+3-4+...+77-78+79-80\)
\(\Rightarrow A=-1-1-...-1-1\) ( \(40\)cặp)
\(\Rightarrow A=-1\times40=-41\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(a;\frac{2^{78}+2^{79}+2^{80}}{2^{77}+2^{76}+2^{75}}=\frac{2^{78}\left(1+2+2^2\right)}{2^{75}\left(1+2+2^2\right)}=2^3=8\)
b) \(\frac{5^{56}+5^7}{5^{49}+1}=\frac{5^7\left(5^{49}+1\right)}{5^{49}+1}=5^7\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(\left(2^{78}+2^{79}+2^{80}\right):\left(2^{77}+2^{76}+2^{75}\right)=\left(2^{78}.7\right):\left(2^{75}.7\right)=2^{78}:2^{75}=8\)
![](https://rs.olm.vn/images/avt/0.png?1311)
9: \(=1-\dfrac{1}{99}+1-\dfrac{1}{100}+\dfrac{100}{101}\cdot\dfrac{1-4+3}{12}=2-\dfrac{199}{9900}=\dfrac{19601}{9900}\)
10: \(=\left(\dfrac{78}{79}+\dfrac{79}{80}+\dfrac{80}{81}\right)\cdot\dfrac{6+5+9-20}{30}=0\)
![](https://rs.olm.vn/images/avt/0.png?1311)
=>(1/2+1/3+...+1/80)*x>(1+1/79+1+2/78+...+1+78/2+1)
=>\(x\left(\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{80}\right)>\dfrac{80}{80}+\dfrac{80}{79}+...+\dfrac{80}{3}+\dfrac{80}{2}\)
=>x>80