\(S=\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{18.19.20}\)
\(2S=\frac{2}{1.2.3}+\frac{2}{2.3.4}+...+\frac{2}{18.19.20}\)
\(=\left(\frac{1}{1.2}-\frac{1}{2.3}\right)+\left(\frac{1}{2.3}-\frac{1}{3.4}\right)+...+\left(\frac{1}{18.19}+\frac{1}{19.20}\right)\)
\(=\frac{1}{2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{18.19}-\frac{1}{19.20}\)
\(=\frac{1}{2}-\frac{1}{19.20}< \frac{1}{2}\)
\(2S< \frac{1}{2}\)
\(\Rightarrow S< \frac{1}{4}\) (ĐPCM)
\(S=\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{18.19.20}\)
\(=\frac{1}{2}\left(\frac{2}{1.2.3}+\frac{2}{2.3.4}+...+\frac{2}{18.19.20}\right)\)
\(=\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{18.19}-\frac{1}{19.20}\right)\)
\(=\frac{1}{2}\left(\frac{1}{2}-\frac{1}{19.20}\right)\)
\(=\frac{1}{2}\left(\frac{1}{2}-\frac{1}{380}\right)\)
\(=\frac{1}{4}-\frac{1}{760}\)
=> S < \(\frac{1}{4}\)( vì 1/4 > 0)
Ta có:\(\frac{1}{n\left(n+1\right)\left(n+2\right)}=\frac{2+n-n}{2n\left(n+1\right)\left(n+2\right)}\)
\(=\frac{1}{2}\left(\frac{2+n-n}{n\left(n+1\right)\left(n+2\right)}\right)\)
\(=\frac{1}{2}\left(\frac{2+n}{n\left(n+1\right)\left(n+2\right)}-\frac{n}{n\left(n+1\right)\left(n+2\right)}\right)\)
\(=\frac{1}{2}\left(\frac{1}{n\left(n+1\right)}-\frac{1}{\left(n+1\right)\left(n+2\right)}\right)\)
Áp dụng:
\(S=\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{18.19}-\frac{1}{19.20}\right)\)
\(=\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{19.20}\right)=\frac{1}{4}-\frac{1}{2.19.20}< \frac{1}{4}\left(dpcm\right)\)
=> 2s=2/1.2.3+1/2.3.4+....+1/18.19.20
=>2s=(3-1)/1.2.3+(4-2)/2.3.4+....+(20-18)/18.19.20
=> 2s=3/1.2.3-1/1.2.3+4/2.3.4-2/2.3.4+...+20/18.19.20-18/18.19.20
=>2s=1/1.2-1/2.3+1/2.3-1/3.4+.....+1/18.19-1/19.20
=> 2s=1/2-1/19.20
=>2s=189/380
=>s=189/760<190/760=1/4
vậy s<1/4
study well
k nha
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