CMR: \(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+....+\frac{1}{\sqrt{225}}< 28\)
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CMR: \(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{225}}< 28\)
\(\frac{1}{\sqrt{2}}=\frac{2}{2\sqrt{2}}< \frac{2}{\sqrt{2}+\sqrt{1}}=\frac{2\left(\sqrt{2}-1\right)}{\left(\sqrt{2}+1\right)\left(\sqrt{2}-1\right)}=2\left(\sqrt{2}-1\right)\)
\(\frac{1}{\sqrt{3}}=\frac{2}{2\sqrt{3}}< \frac{2}{\sqrt{3}+\sqrt{2}}=\frac{2\left(\sqrt{3}-\sqrt{2}\right)}{\left(\sqrt{3}+\sqrt{2}\right)\left(\sqrt{3}-\sqrt{2}\right)}=2\left(\sqrt{3}-\sqrt{2}\right)\)
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\(\frac{1}{\sqrt{225}}=\frac{2}{2\sqrt{225}}< \frac{2}{\sqrt{225}+\sqrt{224}}=\frac{2\left(\sqrt{225}-\sqrt{224}\right)}{\left(\sqrt{225}+\sqrt{224}\right)\left(\sqrt{225}-\sqrt{224}\right)}\)\(=2\left(\sqrt{225}-\sqrt{224}\right)\)
\(\Rightarrow\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{225}}< 2\left(\sqrt{2}-1+\sqrt{3}-\sqrt{2}+...+\sqrt{225}-\sqrt{224}\right)\)
\(\Rightarrow\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{225}}< 2\left(\sqrt{225}-1\right)=2\left(15-1\right)=28\)
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CMR \(\frac{1}{\sqrt{2}}\)+\(\frac{1}{\sqrt{3}}\)+.....+\(\frac{1}{\sqrt{225}}\)<28
Chứng minh : \(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+....+\frac{1}{\sqrt{225}}< 28\)
vế phải < \(2.\left(\frac{1}{2\sqrt{2}}+\frac{1}{2\sqrt{3}}+...+\frac{1}{2\sqrt{225}}\right)\)
<\(2\left(\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+...+\frac{1}{\sqrt{224}+\sqrt{225}}\right)\)
\( =2.\left(-1+\sqrt{2}-\sqrt{2}+\sqrt{3}-...-\sqrt{224}+\sqrt{225}\right)\)
=\(2.\left(-1+\sqrt{225}\right)=2.14=28\)
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Chứng minh rằng:
a) \(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{4}}+...+\frac{1}{\sqrt{225}}>28\)
c/m
\(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{225}}< 28\)
Tính :\(A=\frac{1}{2\sqrt{1}+1\sqrt{2}}+\frac{1}{3\sqrt{2}+2\sqrt{3}}+\frac{1}{4\sqrt{3}+3\sqrt{4}}+....+\frac{1}{225\sqrt{224}+224\sqrt{225}}\)
Sorry mới lớp 6 chưa học
thông cảm
no chửi
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Ta có:
\(\frac{1}{\left(n+1\right)\sqrt{n}+n\sqrt{n+1}}=\frac{1}{\sqrt{n\left(n+1\right)}.\left(\sqrt{n}+\sqrt{n+1}\right)}\)
\(=\frac{1}{\sqrt{n\left(n+1\right)}.\left(\sqrt{n}+\sqrt{n+1}\right)}=\frac{\sqrt{n+1}-\sqrt{n}}{\sqrt{n\left(n+1\right)}}=\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\)
Thế vào bài toán ta được
\(A=\frac{1}{2\sqrt{1}+1\sqrt{2}}+\frac{1}{3\sqrt{2}+2\sqrt{3}}+...+\frac{1}{225\sqrt{224}+224\sqrt{225}}\)
\(=\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{224}}-\frac{1}{\sqrt{225}}\)
\(=1-\frac{1}{\sqrt{225}}=1-\frac{1}{15}=\frac{14}{15}\)
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cm : \(\frac{1}{\sqrt{2}}\)+\(\frac{1}{\sqrt{3}}\)+...\(\frac{1}{\sqrt{225}}\)<28
Tính
\(\frac{1}{2\sqrt{1}+1\sqrt{2}}+\frac{1}{3\sqrt{2}+2\sqrt{3}}+...+\frac{1}{225\sqrt{224}+224\sqrt{225}}\)
1) CMR frac{1}{sqrt{1.1999}}+frac{1}{sqrt{2.1998}}+frac{1}{sqrt{3.1997}}+...+frac{1}{sqrt{1999.1}}ge1,9992) CMR frac{1}{1sqrt{2}+2sqrt{1}}+frac{1}{3sqrt{2}+2sqrt{3}}+...+frac{1}{95sqrt{94}+94sqrt{95}} 13) CMR frac{1}{2sqrt{1}}+frac{1}{3sqrt{2}}+frac{1}{4sqrt{3}}+...+frac{1}{left(n+1right)sqrt{n}} 24) CMR sqrt{n} frac{1}{sqrt{1}}+frac{1}{sqrt{2}}+frac{1}{sqrt{3}}+...+frac{1}{sqrt{n}} 2sqrt{n}
Đọc tiếp
1) CMR \(\frac{1}{\sqrt{1.1999}}+\frac{1}{\sqrt{2.1998}}+\frac{1}{\sqrt{3.1997}}+...+\frac{1}{\sqrt{1999.1}}\ge1,999\)
2) CMR \(\frac{1}{1\sqrt{2}+2\sqrt{1}}+\frac{1}{3\sqrt{2}+2\sqrt{3}}+...+\frac{1}{95\sqrt{94}+94\sqrt{95}}< 1\)
3) CMR \(\frac{1}{2\sqrt{1}}+\frac{1}{3\sqrt{2}}+\frac{1}{4\sqrt{3}}+...+\frac{1}{\left(n+1\right)\sqrt{n}}< 2\)
4) CMR \(\sqrt{n}< \frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{n}}< 2\sqrt{n}\)