a: \(\left(2x-1\right)^2-x\left(4x-3\right)\)

\(=4x^2-4x+1-4x^2+3x\)

=-x+1

b: \(\left(x-2\right)\left(x-1\right)-\left(x-3\right)^2\)

\(=x^2-3x+2-\left(x^2-6x+9\right)\)

\(=x^2-3x+2-x^2+6x-9=3x-7\)

c: \(\left(x-2\right)^3-x\left(x-1\right)\left(x-2\right)\)

\(=x^3-6x^2+12x-8-x\left(x^2-3x+2\right)\)

\(=x^3-6x^2+12x-8-x^3+3x^2-2x\)

\(=-3x^2+10x-8\)

d: \(\frac12x\left(x-2\right)-\left(2x-3\right)^2\)

\(=\frac12x^2-x-\left(4x^2-12x+9\right)\)

\(=\frac12x^2-x-4x^2+12x-9=-\frac72x^2+11x-9\)

\(\frac43+56=\frac43+\frac{168}{3}=\frac{172}{3}\)

\(=\frac43+\frac{56}{1}\)

\(=\frac43+\frac{168}{3}\)

\(=\frac{4+168}{3}\)

\(=\frac{172}{3}\)

Ta có: \(\frac{x}{20}+\frac{x}{30}+\frac{x}{42}=3\)

=>\(x\left(\frac{1}{20}+\frac{1}{30}+\frac{1}{42}\right)=3\)

=>\(x\left(\frac14-\frac15+\frac15-\frac16+\frac16-\frac17\right)=3\)

=>\(x\left(\frac14-\frac17\right)=3\)

=>\(x\cdot\frac{3}{21}=3\)

=>\(x=3:\frac{3}{21}=21\)

\(\frac{x}{20}+\frac{x}{30}+\frac{x}{42}=\) \(3\)

\(\frac{21x}{420}+\frac{14x}{420}+\frac{10x}{420}=3\)

\(\frac{45x}{420}=3\)

\(45x=3\times420\)

\(45x=1260\)

\(x=1260:45\)

\(x=28\)

A = 1.2 + 2.3 + ...+ n(n + 1)

1.2.3 = 1.2.3

2.3.3 = 2.3(4-1) = 2.3.4 - 1.2.3

.............................................................

n(n + 1).3 = n(n + 1).{(n + 2) - (n-1)} = n(n+1)(n+2)-(n-1)n(n+2)

Cộng vế với vế ta có:

3A = n(n+1)(n+2)

A = \(\frac{n\left(n+1\right)\left(n+2\right)}{3}\)

A = 1\(^2\) + \(2^2\) + ...+ n\(^2\)

A = 1 + 2.(1+ 1) + ...+ n[(n - 1) + 1]

A = 1 + 2.1 + 2 + ...+ n(n-1) + n

A = (1 + 2 + ..+n) + [1.2 + 2.3 + 3.4 +...+(n-1)n]

Đặt B = 1 + 2+ .. +n

C = 1.2 + 2.3 +..+ (n -1)n

B = 1 + 2+ ...+ n

B =(n + 1).n : 2

1.2.3 = 1.2.3

2.3.3 = 2.3.(4-1) = 2.3.4 - 1.2.3

3.4.3 = 3.4.(5- 2) = 3.4.5 - 2.3.4

................................................................

(n -1).n.3 = (n - 1).n.[(n +1) - (n - 2)] = (n-1)n(n+1) -(n-2)(n-1)n

Cộng vế với vế ta có:

3B = (n-1)n(n+1)

B = \(\frac{\left(n-1\right)n\left(n+1\right)}{3}\)

A = B + C

A = \(\frac{n\left(n+1\right)}{2}\) + \(\frac{\left(n-1\right)n\left(n+1\right)}{3}\)

A = n(n+1).(\(\frac12\) + \(\frac{n-1}{3}\))

A = n(n+1).(\(\frac{3+2n-2}{6}\))

A = n(n+1).\(\frac{2n+1}{6}\)

A =\(\frac{n\left(n+1\right)\left(2n+1\right)}{6}\)

- \(\frac{15}{8}\) - \(\frac{23}{12}\) + \(\frac53\) - (\(\frac{25}{12}\) + \(\frac{-7}{8}\))

= - \(\frac{15}{8}\) - \(\frac{23}{12}\) + \(\frac53\) - \(\frac{25}{12}\) + \(\frac78\)

= -(\(\frac{15}{8}\) - \(\frac78\)) - (\(\frac{23}{12}\) + \(\frac{25}{12}\)) + \(\frac53\)

= - 1 - 4 + \(\frac53\)

= - 5 + \(\frac53\)

= - \(\frac{15}{3}\) + \(\frac53\)

= - \(\frac{10}{3}\)

4 : \(\frac{5}{21}\) = 4 x \(\frac{21}{5}\) = \(\frac{48}{5}\)

4 : \(\frac{5}{21}\)

= 4 . \(\frac{21}{5}\)

= \(\frac{84}{5}\)

Chúc bạn học tốt!

\(1,1\left(9\right)=1,1+0,0\left(9\right)=\frac{11}{10}+\frac{1}{10}=\frac{12}{10}=\frac65\)

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\(\frac{2}{1\cdot2}+\frac{2}{2\cdot3}+\cdots+\frac{2}{19\cdot20}\)

\(=2\left(\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\cdots+\frac{1}{19\cdot20}\right)\)

\(=2\left(1-\frac12+\frac12-\frac13+\cdots+\frac{1}{19}-\frac{1}{20}\right)\)

\(=2\left(1-\frac{1}{20}\right)=2\cdot\frac{19}{20}=\frac{19}{10}\)

Đặt \(A=\frac{2}{1\times2}+\frac{2}{2\times3}+\cdots+\frac{2}{18\times19}+\frac{2}{19\times20}\)

Ta có:

\(A=2\times\left(\frac{1}{1\times2}+\frac{1}{2\times3}+\cdots+\frac{1}{18\times19}+\frac{1}{19\times20}\right)\)

\(A=2\times\left(\frac{2-1}{1\times2}+\frac{3-2}{2\times3}+\cdots+\frac{19-18}{18\times19}+\frac{20-19}{19\times20}\right)\)

\(A=2\times\left(\frac{2}{1\times2}-\frac{1}{1\times2}+\frac{3}{2\times3}-\frac{2}{2\times3}+\cdots+\frac{20}{19\times20}-\frac{19}{19\times20}\right)\)

\(A=2\times\left(1-\frac12+\frac12-\frac13+\ldots+\frac{1}{19}-\frac{1}{20}\right)\)

\(A=2\times\left(1-\frac{1}{20}\right)\)

\(A=2\times\frac{19}{20}\)

\(A=\frac{19}{10}\)

Vậy \(A=\frac{19}{10}\)