\(C=\dfrac{5}{28}+\dfrac{1}{14}+\dfrac{1}{26}+...+\dfrac{1}{638}\)

\(=\dfrac{5}{28}+\dfrac{5}{70}+\dfrac{5}{130}+...+\dfrac{5}{3190}\)

\(=\dfrac{5}{4\cdot7}+\dfrac{5}{7\cdot10}+...+\dfrac{5}{55\cdot58}\)

\(=\dfrac{5}{3}\left(\dfrac{3}{4\cdot7}+\dfrac{3}{7\cdot10}+...+\dfrac{3}{55\cdot58}\right)\)

\(=\dfrac{5}{3}\left(\dfrac{1}{4}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{10}+...+\dfrac{1}{55}-\dfrac{1}{58}\right)\)

\(=\dfrac{5}{3}\left(\dfrac{1}{4}-\dfrac{1}{58}\right)=\dfrac{5}{3}\cdot\dfrac{27}{116}=\dfrac{5\cdot9}{116}=\dfrac{45}{116}\)

\(C=\dfrac{5}{28}+\dfrac{1}{14}+\dfrac{1}{26}+...+\dfrac{1}{638}\\ =\dfrac{5}{28}+\dfrac{5}{70}+\dfrac{5}{130}+...+\dfrac{5}{3190}\\ =\dfrac{5}{4\cdot7}+\dfrac{5}{7\cdot10}+\dfrac{5}{10\cdot13}+...+\dfrac{5}{55\cdot58}\\ =\dfrac{5}{3}\cdot\left(\dfrac{3}{4\cdot7}+\dfrac{3}{7\cdot10}+\dfrac{3}{10\cdot13}+...+\dfrac{3}{55\cdot58}\right)\\ =\dfrac{5}{3}.\left(\dfrac{1}{4}-\dfrac{1}{7}+\dfrac{1}{7}-...-\dfrac{1}{55}+\dfrac{1}{55}-\dfrac{1}{58}\right)\\ =\dfrac{5}{3}\cdot\left(\dfrac{1}{4}-\dfrac{1}{58}\right)\\ =\dfrac{5}{3}\cdot\dfrac{27}{116}\\ =\dfrac{45}{116}\)

Số số hạng của dãy số A là:

\(\dfrac{199-1}{2}+1=\dfrac{198}{2}+1=100\left(số\right)\)

Tổng của dãy số A là:

\(\left(1+199\right)\cdot\dfrac{100}{2}=100^2=10000\)

Số số hạng của dãy số B là:

\(\dfrac{999-100}{1}+1=899+1=900\left(số\right)\)

Tổng của dãy số là: \(B=\left(999+100\right)\cdot\dfrac{900}{2}=494550\)

\(A+B=10000+494550=504550\)

cc

\(2+2^2+2^3+2^4+...+2^9+2^{10}\)

\(=2\left(1+2\right)+2^3\left(1+2\right)+...+2^9\left(1+2\right)\)

\(=3\left(2+2^3+...+2^9\right)⋮3\)

\(3^x-3^{x-1}=54\)

=>\(3^x-\dfrac{1}{3}\cdot3^x=54\)

=>\(\dfrac{2}{3}\cdot3^x=54\)

=>\(3^x=54:\dfrac{2}{3}=81=3^4\)

=>x=4

a: \(2^5\cdot4^3:8^2=2^5\cdot2^6:2^6=2^5\)

b: \(3^9:27^2\cdot243=3^9:3^6\cdot3^5=3^8\)

`4^8*2^20`

`=(2^2)^8*2^20`

`=2^(2*8)*2^20`

`=2^16*2^20`

`=2^(16+20)`

`=2^36` 

\(4^8\cdot2^{20}=\left(2^2\right)^8\cdot2^{20}=2^{16}\cdot2^{20}=2^{16+20}=2^{36}\)

`a)3^9*27-3^7`

`=3^9:3^3-3^7`

`=3^(9-3)-3^7`

`=3^6-3^7`

`=3^6*(1-3)`

`=-2*3^6`

`=-1458` 

`b)(3^8+3^7):3^6`

`=3^8:3^6+3^7:3^6`

`=3^(8-6)+3^(7-6)`

`=3^2+3`

`=9+3`

`=12` 

`a)3+3^2+3^3+3^4`

`=(3+3^2)+(3^3+3^4)` 

`=3*(1+3)+3^3*(1+3)`

`=4*3+4*3^3`

`=4*(3+3^3)` chia hết cho 4

`b)5+5^2+...+5^20`

`=(5+5^2)+(5^3+5^4)+...+(5^19+5^20)`

`=5*(1+5)+5^3*(1+5)+...+5^19*(1+5)`

=6*5+6*5^3+...+6*5^19`

`=6*(5+5^3+...+5^19)` chia hết cho 6 

\(a,4^x\cdot13=832\\ 4^x=832:13\\ 4^x=64\\ 4^x=4^3\\ x=3\\ b,\left(2x-1\right)^3=5^2\cdot2^2+25\\ \left(2x-1\right)^3=25\cdot4+25\\ \left(2x-1\right)^3=125\\ \left(2x-1\right)^3=5^3\\ 2x-1=5\\ 2x=5+1\\ 2x=6\\ x=3\)