B= 112+122+133+....+1992<11.2+12.3+...+199.100112+122+133+....+1992<11.2+12.3+...+199.100
Ta có: 11.2+12.3+13.4+...+199.10011.2+12.3+13.4+...+199.100
= 1−12+12−13+13−.....−199+199−11001−12+12−13+13−.....−199+199−1100
= 1−1100=99100<1<1341−1100=99100<1<134
Vậy B < 134134.
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B = \(\dfrac{1}{1^{2^{ }}}+\dfrac{1}{2^2}+\dfrac{1}{3^3}+....+\dfrac{1}{99^2}< \dfrac{1}{1.2}+\dfrac{1}{2.3}+...+\dfrac{1}{99.100}\)
Ta có: \(\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{99.100}\)
= \(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-.....-\dfrac{1}{99}+\dfrac{1}{99}-\dfrac{1}{100}\)
= \(1-\dfrac{1}{100}=\dfrac{99}{100}< 1< 1\dfrac{3}{4}\)
Vậy B < \(1\dfrac{3}{4}\).
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