tại sao 1/2 - 1/81 > 1/2

Chuẩn chuẩn. :)

\(A=\frac{1}{4^2}+\frac{1}{6^2}+...+\frac{1}{160^2}=\frac{1}{4}\left(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{80^2}\right)\)

+) \(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{80^2}>\frac{1}{4}+\left(\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{80}.\frac{1}{81}\right)\)

\(=\frac{1}{4}+\left(\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{80}-\frac{1}{81}\right)\)

\(=\frac{1}{4}+\frac{1}{3}-\frac{1}{81}>\frac{1}{4}+\frac{1}{3}-\frac{1}{12}=\frac{1}{2}\)

=> \(A=\frac{1}{4}\left(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{80^2}\right)>\frac{1}{4}.\frac{1}{2}=\frac{1}{8}\)

+) \(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{80^2}< \frac{1}{4}+\left(\frac{1}{3.2}+\frac{1}{4.3}+...+\frac{1}{80.79}\right)\)

\(=\frac{1}{4}+\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{79}-\frac{1}{80}\right)\)

\(=\frac{1}{4}+\frac{1}{2}-\frac{1}{80}< \frac{3}{4}\)

=> \(A=\frac{1}{4}\left(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{80^2}\right)< \frac{1}{4}.\frac{3}{4}=\frac{3}{16}\)

Bài làm

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

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

= \(\frac{1}{2}.\frac{2}{3}.\frac{3}{4}.....\frac{99}{100}\)

= \(\frac{1.2.3.....99}{2.3.4.....100}\)

= \(\frac{1.1.1.....1}{1.1.1.....100}\)

= \(\frac{1}{100}\)

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có bài nào khó thì bạn giải giúp mk nhé

\(A=1+2\left(1+1\right)+3\left(2+1\right)+4\left(3+1\right)+...+100.\left(99+1\right).\)

\(A=1+1.2+2+2.3+3+3.4+4+...+99.100+100\)

\(A=\left(1+2+3+4+...+100\right)+\left(1.2+2.3+3.4+...+99.100\right)\)

\(B=1+2+3+4+...+100=\frac{100\left(1+100\right)}{2}=5050\)

\(C=1.2+2.3+3.4+...+99.100\)

\(3C=1.2.3+2.3.3+3.4.3+...+99.100.3\)

\(3C=1.2.3+2.3\left(4-1\right)+3.4\left(5-2\right)+...+99.100\left(101-98\right)\)

\(3C=1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+...+99.100.101-98.99.100\)

\(3C=99.100.101\Rightarrow C=\frac{99.100.101}{3}=33.100.101=333300\)

\(A=B+C=5050+333300=338350\)

ai gải giúp mk nhanh lên đi

rồi mk tk cho

2013/2014