M=1.2+2.3+3.4+...+19.20

3.M=1.2.3+2.3.3+3.4.3+...+19.20.3

3.M=1.2.(3-0)+2.3.(4-1)+3.4.(5-2)+...+19.20.(21-18)

3.M=(1.2.3-0.1.2)+(2.3.4-1.2.3)+(3.4.5-2.3.4)+...+(19.20.21-18.19.20)

Những cái bị gạch là giản ước.

3.M=19.20.21-0.1.2

3.M=7980-0

3.M=7980

M=7980:3

M=2660

Vậy M=2660

Dấu . là dấu nhân

2432902008176640000

\(C=1.2+2.3+3.4+...+x.\left(x-1\right)\)

\(\Rightarrow3C=1.2.3+2.3.3+3.4.3+...+x.\left(x-1\right).3\)

\(\Rightarrow3C=1.2.\left(3-0\right)+2.3.\left(4-1\right)+3.4.\left(5-2\right)+...+x.\left(x-1\right).\left[\left(x+1\right)-\left(x-2\right)\right]\)

\(\Rightarrow3C=\left(1.2.3-0.12\right)+\left(2.3.4-1.2.3\right)+\left(3.4.5-2.3.4\right)+...+\left[x.\left(x-1\right)\left(x+1\right)-x.\left(x-1\right)\left(x-2\right)\right]\)

\(\Rightarrow3C=-0.1.2+x.\left(x-1\right)\left(x+1\right)\)

\(\Rightarrow3C=x.\left(x-1\right)\left(x+1\right)\)

\(\Rightarrow C=\dfrac{x.\left(x-1\right)\left(x+1\right)}{3}\)

3C=1x2x3+2x3x3+3x4x3+...+Xx(X+1)=

=1x2x3+2x3x(4-1)+3x4x(5-2)+...+Xx(X+1)[(X+2)-(X-1)]=

=1x2x3-1x2x3+2x3x4-2x3x4+3x4x5-...-(X-1)xXx(X+1)+Xx(X+1)x(X+2)=

=Xx(X+1)(X+2)

Bài 1:

Đặt \(A=\frac{2}{1x2}+\frac{2}{2x3}+\frac{2}{3x4}+...+\frac{2}{18x19}+\frac{2}{19x20}\)

\(\frac{A}{2}=\frac{1}{1x2}+\frac{1}{2x3}+\frac{1}{3x4}+...+\frac{1}{18x19}+\frac{1}{19x20}\)

\(\frac{A}{2}=\frac{2-1}{1x2}+\frac{3-2}{2x3}+\frac{4-3}{3x4}+...+\frac{19-18}{18x19}+\frac{20-19}{19x20}\)

\(\frac{A}{2}=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{18}-\frac{1}{19}+\frac{1}{19}-\frac{1}{20}=1-\frac{1}{20}=\frac{19}{20}\)

\(A=\frac{2x19}{20}=\frac{19}{10}\)

Bài 2:

Đặt \(B=\frac{1}{1x2}+\frac{1}{2x3}+\frac{1}{3x4}+...+\frac{1}{8x9}+\frac{1}{9x10}\)

Làm tương tự câu 1 có \(B=1-\frac{1}{10}=\frac{9}{10}\)

\(Bx100=\frac{9}{10}x100=90\)

=> \(\left[\frac{5}{2}:\left(x+\frac{206}{100}\right)\right]:\frac{1}{2}=1\)

=> \(\left[\frac{5}{2}:\left(x+\frac{206}{100}\right)\right]=\frac{1}{2}\)

=>  \(x+\frac{206}{100}=\frac{5}{2}:\frac{1}{2}=5\Rightarrow x=5-\frac{206}{100}=\frac{294}{100}=\frac{147}{50}\)

bài 1 đáp án là:19/10

2:147/50

đáp án bài 1 là: 19/10

đáp án bài 2 là 147/50

A=1²/1.2 .2²/2.3 .3²/3.4 .4²/4.5

=1/2. 2.2/2.3 .3.3/3.4 .4.4/4.5

=1/2.2/3.3.4.4./5

=1/5

Đặt A = 1×2 + 2×3 + 3×4 + ... + 19×20

⇒ 3A = 1×2×3 + 2×3×3 + 3×4×3 + ... + 19×20×3

= 1×2×3 + 2×3×(4 - 1) + 3×4×(5 - 2) + ... + 19×20×(21 - 18)

= 1×2×3 - 1×2×3 + 2×3×4 - 2×3×4 + 3×4×5 - ... - 18×19×20 + 19×20×21

= 19×20×21

= 7980

⇒ A = 7980 : 3 = 2660

\(A=1\times2+2\times3+3\times4+...+19\times20\)

\(A\times3=3\times\left(1\times2+2\times3+3\times4+...+19\times20\right)\)

\(A\times3=1\times2\times3+2\times3\times3+3\times4\times3+...+19\times20\times3\)

\(A\times3=1\times2\times3+2\times3\times\left(4-1\right)+3\times4\times\left(5-2\right)+....+19\times20\times\left(21-18\right)\)

\(A\times3=1\times2\times3-1\times2\times3+2\times3\times4-2\times3\times4+3\times4\times5+...+19\times20\times21\)

\(A\times3=\left(1\times2\times3-1\times2\times3\right)+\left(2\times3\times4-2\times3\times4\right)+...+\left(18\times19\times20-18\times19\times20\right)+19\times20\times21\)

\(A\times3=19\times20\times21\)

\(A\times3=7980\)

cảm ơn bạn nhiều nhé

\(A=\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{14.15}\)

\(A=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{14}-\frac{1}{15}\)

\(A=1-\frac{1}{15}\)

\(A=\frac{14}{ }15\)

=1-1/2+1/2-1/3+...+1/14-1/15

=1-1/15

=14/15

\(\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{14.15}=\left(1-\frac{1}{2}\right)+\left(\frac{1}{2}-\frac{1}{3}\right)+...+\left(\frac{1}{14}-\frac{1}{15}\right).\)

\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-...-\frac{1}{14}+\frac{1}{14}-\frac{1}{15}=1-\left(\frac{1}{2}-\frac{1}{2}\right)_{ }-...-\left(\frac{1}{14}-\frac{1}{14}\right)-\frac{1}{15}_{ }.\)

\(=1-\frac{1}{15}=\frac{14}{15}.\)