Câu 1:

a: \(A=\frac23+\frac49+\frac27-\frac75+\frac{1}{21}-\frac{2}{45}+4\frac{2015}{2016}\)

\(=\left(\frac23+\frac27+\frac{1}{21}\right)+\left(\frac49-\frac75-\frac{2}{45}\right)+4+\frac{2015}{2016}\)

\(=\left(\frac{14}{21}+\frac{6}{21}+\frac{1}{21}\right)+\left(\frac{20}{45}-\frac{63}{45}-\frac{2}{45}\right)+4+\frac{2015}{2016}\)

\(=1-1+4+\frac{2015}{2016}=4\frac{2015}{2016}\)

b: \(B=\frac{6}{15\cdot18}+\frac{6}{18\cdot21}+\cdots+\frac{6}{87\cdot90}\)

\(=2\left(\frac{3}{15\cdot18}+\frac{3}{18\cdot21}+\cdots+\frac{3}{87\cdot90}\right)\)

\(=2\left(\frac{1}{15}-\frac{1}{18}+\frac{1}{18}-\frac{1}{21}+\cdots+\frac{1}{87}-\frac{1}{90}\right)\)

\(=2\left(\frac{1}{15}-\frac{1}{90}\right)=2\cdot\frac{5}{90}=\frac{10}{90}=\frac19\)

c: \(C=\frac{20\cdot2^4+12\cdot2^4-48\cdot2^2}{8^2}\)

\(=\frac{2^4\cdot\left(20+12\right)-2^4\cdot3\cdot2^2}{2^6}=\frac{2^4\cdot32-2^6\cdot3}{2^6}\)

\(=\frac{2^6\cdot8-2^6\cdot3}{2^6}=8-3=5\)

d: Số số hạng của dãy là:

(173-1):4+1=172:4+1=43+1=44(số)

\(D=1-5+9-13+\cdots+169-173\)

=(1-5)+(9-13)+...+(169-173)

=(-4)+(-4)+...+(-4)

\(=\left(-4\right)\cdot\frac{44}{2}=-4\cdot22=-88\)

e: \(E=1\cdot2+2\cdot3+\cdots+216\cdot217\)

\(=1\left(1+1\right)+2\left(2+1\right)+\cdots+216\left(216+1\right)\)

\(=\left(1^2+2^2+\cdots+216^2\right)+\left(1+2+\cdots+216\right)\)

\(=\frac{216\left(216+1\right)\left(2\cdot216+1\right)}{6}+\frac{216\cdot217}{2}\)

\(=36\cdot217\cdot433+108\cdot217=3406032\)

f: \(G=\frac{3}{2^2}\cdot\frac{8}{3^2}\cdot\ldots\cdot\frac{2499}{50^2}\)

\(=\frac{2^2-1}{2^2}\cdot\frac{3^2-1}{3^2}\cdot\ldots\cdot\frac{50^2-1}{50^2}\)

\(=\frac{\left(2-1\right)\left(2+1\right)}{2^2}\cdot\frac{\left(3-1\right)\left(3+1\right)}{3^2}\cdot\ldots\cdot\frac{\left(50-1\right)\left(50+1\right)}{50^2}\)

\(=\frac{1\cdot2\cdot\ldots\cdot49}{2\cdot3\cdot\ldots\cdot50}\cdot\frac{3\cdot4\cdot\ldots\cdot51}{2\cdot3\cdot\ldots\cdot50}=\frac{1}{50}\cdot\frac{51}{2}=\frac{51}{100}\)

g: \(H=\frac{5\cdot4^{15}\cdot9^9-4\cdot8^9\cdot3^{20}}{5\cdot2^9\cdot6^{19}-7\cdot2^{29}\cdot27^6}\)

\(=\frac{5\cdot2^{30}\cdot3^{18}-2^2\cdot2^{27}\cdot3^{20}}{5\cdot2^9\cdot2^{19}\cdot3^{19}-7\cdot2^{29}\cdot3^{18}}=\frac{5\cdot2^{30}\cdot3^{18}-2^{29}\cdot3^{20}}{5\cdot2^{28}\cdot3^{19}-7\cdot2^{29}\cdot3^{18}}\)

\(=\frac{2^{29}\cdot3^{18}\left(5\cdot2-3^2\right)}{2^{28}\cdot3^{18}\left(5\cdot3-7\cdot2\right)}=2\)

Câu 2:

a: \(\left|2x+1\right|+\left|-\frac12\right|=3,5\)

=>|2x+1|=3,5-0,5=3

=>\(\left[\begin{array}{l}2x+1=3\\ 2x+1=-3\end{array}\right.\Rightarrow\left[\begin{array}{l}2x=2\\ 2x=-4\end{array}\right.\Rightarrow\left[\begin{array}{l}x=1\\ x=-2\end{array}\right.\)

b: \(4^{3x+1}+8^{2x+1}=12\cdot2^{2016}\)

=>\(2^{6x+2}+2^{6x+3}=2^{2018}\cdot3\)

=>\(2^{6x+2}\cdot3=2^{2018}\cdot3\)

=>6x+2=2018

=>6x=2016

=>x=336

d: 5(2x-4)-3(x+1)=-2(x-9)

=>10x-20-3x-3=-2x+18

=>7x-23=-2x+18

=>9x=41

=>x=41/9

h:20 là bội của 2x-1

mà 2x-1 lẻ

nên 2x-1∈{1;-1;5;-5}

=>2x∈{2;0;6;-4}

=>x∈{1;0;3;-2}