1a.
\(A=\frac{2}{3}+\frac{4}{9}+\frac{2}{7}-\frac{7}{5}+\frac{1}{21}-\frac{2}{45}+4\frac{2015}{2016}\)
\(=(\frac{2}{3}+\frac{2}{7}+\frac{1}{21})+(\frac{4}{9}-\frac{7}{5}-\frac{2}{45})+4\frac{2015}{2016}\)
\(=\frac{14+6+1}{21}+\frac{20-63-2}{45}+4\frac{2015}{2016}=1-1+4\frac{2015}{2016}=4\frac{2015}{2016}\)
1b.
\(B=\frac{6}{15.18}+\frac{6}{18.21}+\frac{6}{21.24}+....+\frac{6}{87.90}=2(\frac{3}{15.18}+\frac{3}{18.21}+\frac{3}{21.24}+....+\frac{3}{87.90})\)
\(=2\left(\frac{18-15}{15.18}+\frac{21-18}{18.21}+\frac{24-21}{21.24}+....+\frac{90-87}{87.90}\right)\)
\(=2\left(\frac{1}{15}-\frac{1}{18}+\frac{1}{18}-\frac{1}{21}+....+\frac{1}{87}-\frac{1}{90}\right)=2(\frac{1}{15}-\frac{1}{90})=\frac{1}{9}\)
1c.
\(C=\frac{20.2^4+12.2^4-48.2^2}{8^2}=\frac{20.2^4+12.4.2^2-48.2^2}{8^2}=\frac{20.2^4+48.2^2-48.2^2}{8^2}=\frac{20.2^4}{8^2}=5\)
1d.
\(D=1-5+9-13+17-21+....+169-173\)
\(=(1-5)+(9-13)+(17-21)+...+(169-173)\)
\(=\underbrace{(-4)+(-4)+(-4)+....+(-4)}_{22}=(-4).22=-88\)
1e.
\(E=1.2+2.3+3.4+4.5+...+216.217\)
\(3E=1.2.3.+2.3.3+3.4.3+4.5.3+....+216.217.3\)
\(=1.2.3+2.3(4-1)+3.4(5-2)+4.5(6-3)+...+216.217(218-215)\)
\(=(1.2.3+2.3.4+3.4.5+...+216.217.218)-(1.2.3+2.3.4+3.4.5+...+215.216.217)\)
\(=216.217.2018\)
\(\Rightarrow E=72.217.218\)
1f.
\(G=\frac{3}{2^2}.\frac{8}{3^2}.\frac{15}{4^2}....\frac{2499}{50^2}\)
\(=\frac{1.3}{2^2}.\frac{2.4}{3^2}.\frac{3.5}{4^2}....\frac{49.51}{50^2}=\frac{(1.3)(2.4)(3.5)....(49.51)}{(2.3.4...50)^2}\)
\(=\frac{(1.2.3...49)(3.4.5...51)}{(2.3.4...50)(2.3.4....50)}=\frac{1}{50}.\frac{51}{2}=\frac{51}{100}\)
1g.
\(H=\frac{5.4^{15}.9^9-4.8^9.3^{20}}{5.2^9.6^{19}-7.2^{29}.27^6}=\frac{5.2^{30}.3^{18}-2^{29}.3^{20}}{5.2^{28}.3^{19}-7.2^{29}.3^{18}}\) \(=\frac{2^{29}.3^{18}(5.2-3^2)}{2^{28}.3^{18}(5.3-7.2)}=\frac{2^{29}.3^{18}}{2^{28}.3^{18}}=2\)
2a.
\(|2x+1|+|\frac{-1}{2}|=3,5\)
\(\Leftrightarrow |2x+1|+0,5=3,5\)
$\Leftrightarrow |2x+1|=3$\(\Rightarrow \left[\begin{matrix} 2x+1=3\\ 2x+1=-3\end{matrix}\right.\Rightarrow \left[\begin{matrix} x=1\\ x=-2\end{matrix}\right.\)
2b.
\(4^{3x+1}+8^{2x+1}=12.2^{2016}\)
\(\Leftrightarrow 2^{6x+2}+2^{6x+3}=12.2^{2016}\)
\(\Leftrightarrow 2^{6x+2}(1+2)=12.2^{2016}\Leftrightarrow 2^{6x+2}=2^{2018}\)
\(\Rightarrow 6x+2=2018\Rightarrow x=336\)
2c.
\(9^{3x+2}+27^{2x+1}=12.3^{2008}\)
\(\Leftrightarrow 3^{6x+4}+3^{6x+3}=12.3^{2008}\)
\(\Leftrightarrow 3^{6x+3}(3+1)=12.3^{2008}\)
\(\Leftrightarrow 3^{6x+3}=3^{2009}\)
\(\Rightarrow 6x+3=2009\Rightarrow x=\frac{1003}{3}\)
2d.
$5(2x-4)-3(x+1)=-2(x-9)$$\Leftrightarrow 10x-20-3x-3=-2x+18$
$\Leftrightarrow 7x-23=-2x+18$
$\Leftrightarrow 9x=41\Rightarrow x=\frac{41}{9}$
2e.
$|2x-1|+1=x$
$\Rightarrow |2x-1|=x-1$. Vì $|2x-1|\geq 0\Rightarrow x-1\geq 0\Rightarrow x\geq 1$
$\Rightarrow 2x-1>0\Rightarrow |2x-1|=2x-1$. Do đó:
$2x-1+1=x$
$\Leftrightarrow 2x=x\Leftrightarrow x=0$ (vô lý vì $x\geq 1$)
Vậy...............
2g.
\(\frac{1}{3}-|\frac{5}{4}-2x|=\frac{1}{4}\Leftrightarrow |\frac{5}{4}-2x|=\frac{1}{3}-\frac{1}{4}=\frac{1}{12}\)
\(\Rightarrow \left[\begin{matrix} \frac{5}{4}-2x=\frac{1}{12}\\ \frac{5}{4}-2x=-\frac{1}{12}\end{matrix}\right.\Rightarrow \left[\begin{matrix} x=\frac{7}{12}\\ x=\frac{2}{3}\end{matrix}\right.\)
2h.
Với $x$ nguyên thì $2x-1$ là 1 số lẻ (1)
$20$ là bội của $2x-1$, tức là $2x-1$ là ước của $20$ (2)
Từ $(1);(2)$ suy ra $2x-1\left\{\pm 1; \pm 5\right\}$
$\Rightarrow x\in\left\{0; 1; 3; -2\right\}$
2i.
$5^x.5^{x+1}.5^{x+2}\leq 10^{18}:2^{18}$
$\Leftrightarrow 5^{x+x+1+x+2}\leq 5^{18}$
$\Leftrightarrow 5^{3x+3}\leq 5^{18}$
$\Rightarrow 3x+3\leq 18$
$\Rightarrow x\leq 5$
Vì $x$ là số tự nhiên nên $x\in\left\{0;1;2;3;4;5\right\}$
