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Bài 1 :\(a,=\frac{4}{1.3}.\frac{9}{2.4}.\frac{16}{3.5}...\frac{100^2}{99.101}\)
\(=\frac{2.3.4...100}{1.2.3...99}.\frac{2.3.4...100}{3.4...101}\)
\(=100.\frac{2}{101}=\frac{200}{101}\)
Lời giải:
$x(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+\frac{1}{5.6}+\frac{1}{6.7})< 1\frac{6}{7}$
$x(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+\frac{1}{6}-\frac{1}{7})< \frac{13}{7}$
$x(1-\frac{1}{7})< \frac{13}{7}$
$x.\frac{6}{7}< \frac{13}{7}$
$x< \frac{13}{7}: \frac{6}{7}=\frac{13}{6}$
Vì $x$ là số nguyên nên $x\leq 2$
Vậy $x$ là các số nguyên sao cho $x\leq 2$.
a) \(\left(-\frac{5}{2}\right)^2:\left(-15\right)-\left(-0,45+\frac{3}{4}\right).\left(-1\frac{5}{9}\right)\)
= \(-\frac{25}{4}:\left(-15\right)-\left(\frac{9}{20}+\frac{15}{20}\right).\left(-\frac{14}{9}\right)\)
=\(-\frac{25}{4}.\frac{1}{-15}-\frac{6}{5}.\left(-\frac{14}{9}\right)\)
= \(\frac{-5}{12}-\frac{8}{5}\)
= \(\frac{\left(-25\right)-96}{60}\)
= \(\frac{\left(-25\right)+\left(-96\right)}{60}\)
=\(\frac{121}{60}\)
b) \(\left(\frac{-1}{3}\right)-\left(\frac{-3}{5}\right)^0+\left(1-\frac{1}{2}\right)^2:2\)
= \(\left(\frac{-1}{3}\right)-1+\left(\frac{1}{2}\right)^2.\frac{1}{2}\)
=\(\left(\frac{-1}{3}\right)-\frac{3}{3}+\frac{1}{4}.\frac{1}{2}\)
= \(\frac{-4}{3}+\frac{1}{8}\)=\(\frac{-32+3}{24}\)
=\(\frac{-29}{24}\)
c) E=\(\frac{4^5.9^4-2.6^9}{2^{10}.3^8+6^8.20}\)
=\(\frac{\left(2^2\right)^5.\left(3^2\right)^4-2.6^9}{2^{10}.3^8+6^8.20}\)
=\(\frac{2^{10}.3^8-2.6^9}{2^{10}.3^8+6^8.20}\)
=\(\frac{3}{5}\)
d)\(\frac{5^4.20^4}{25^5.4^5}\)
=\(\frac{\left(5.20\right)^4}{\left(25.4\right)^5}\)
=\(\frac{100^4}{100^5}\)
=\(\frac{1}{100}\)
\(\left(\frac{1}{2}-\frac{1}{3}\right).6^{x+1}+6^{x+1}=7.6^9\)
\(\Rightarrow\frac{1}{6}.6.6^x+6.6^x=7.6^9\)
\(\Rightarrow6^x+6.6^x=7.6^9\)
\(\Rightarrow6^x.\left(1+6\right)=7.6^9\)
\(\Rightarrow6^x=\frac{7.6^9}{7}=6^9\)
\(\Rightarrow x=9\)
\(\left(\frac{1}{2}-\frac{1}{3}\right).6^{x+1}+6^{x+1}=7.6^9\)
\(\Leftrightarrow\frac{1}{6}.6^{x+1}+6^{x+1}=7.6^9\)
\(\Leftrightarrow6^{x+1}.\left(\frac{1}{6}+1\right)=7.6^9\)
\(\Leftrightarrow6^{x+1}.\frac{7}{6}=7.6^9\)
\(\Leftrightarrow6^{x+1}=7.6^9:\frac{7}{6}\)
\(\Leftrightarrow6^{x+1}=7.6^9.\frac{6}{7}\)
\(\Leftrightarrow6^{x+1}=\left(7.\frac{6}{7}\right).6^9\)
\(\Leftrightarrow6^{x+1}=6.6^9\)
\(\Leftrightarrow6^{x+1}=6^{10}\)
\(\Leftrightarrow x+1=10\)
\(\Leftrightarrow x=9\)