\(A=\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{49.50}\)

\(=\frac{2-1}{1.2}+\frac{3-2}{2.3}+\frac{4-3}{3.4}+...+\frac{50-49}{49.50}\)

\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{49}-\frac{1}{50}\)

\(=1-\frac{1}{50}=\frac{49}{50}\)

\(B=1.2+2.3+3.4+...+49.50\)

\(3B=1.2.3+2.3.3+3.4.3+...+49.50.3\)

\(=1.2.3+2.3.\left(4-1\right)+3.4.\left(5-2\right)+...+49.50.\left(51-48\right)\)

\(=1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+...+49.50.51-48.49.50\)

\(=49.50.51\)

\(B=\frac{49.50.51}{3}=49.50.17\)

\(50^2.A-\frac{B}{17}=49.50-49.50=0\)

=> 3A = 1.2.3 + 2.3.3 + 3.4.3 + .... + 49.50.3

=> 3A = 1.2.3 + 2.3.( 4 - 1 ) + 3.4.( 5 - 2 ) + .... + 49.50.( 51 - 48 )

=> 3A = 1.2.3 + 2.3.4 - 1.2.3 + 3.4.5 - 2.3.4 + .... + 49.50.51 - 48.49.50

=> 3A = ( 1.2.3 - 1.2.3 ) + ( 2.3.4 - 2.3.4 ) + .... + ( 48.49.50 - 48.49.50 ) + 49.50.51

=> 3A = 49.50.51

=> A = ( 49.50.51 ) : 3 

=> A = 41650

A = 1.2 + 2.3 + 3.4 + ..... + 49.50

3A=1.2.3+2.3.3+3.4.3+...+49.50.3

3A=1.2.3+2.3.(4-1)+3.4.(5-2)+...+48.49.(50-47)+49.50.(51-48)

3A=1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+...+48.49.50-47.48.49+49.50.51-48.49.50

3A=(1.2.3-1.2.3)+(2.3.4-2.3.4)+...(47.48.49-47.48.49)-(48.49.50-48.49.50)+49.50.51

3A=0+0+...+0+0+49.50.51

3A=49.50.51

A=\(\frac{49.50.51}{3}\)

A=41650

Đáp số: A=41650

Ta có : S = 1.2 + 2.3 + 3.4 + ..... + 32.33

=> 3S = 1.2.3 - 1.2.3 + 2.3.4 - 2.3.4 + ...... + 32.33.34

=> 3S = 32.33.34

=> S = \(\frac{32.33.34}{3}=11968\)

\(\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{49.50}\)

= \(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{49}-\frac{1}{50}\)

= \(1-\frac{1}{50}

Ta có : 1/1.2 + 1/2.3 + 1/3.4 + ... + 1/49.50 

= 1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + ... + 1/49 - 1/50

= 1 - 1/50 < 1

Nên 1/1.2 + 1/2.3 + 1/3.4 + ... + 1/49.50 < 1

1/1.2+1/2.3+1/3.4+...+1/49.50

1-1/2+1/2-1/3+/13-1/4+1/4-1/5+1/5-...-1/49+1/49-1/50

1-1/50

50/50-1/50=49/50

E=1/1*2+1/2*3+1/3*4+...+1/49*50

E=1/1-1/2+1/2-1/3+1/3-1/4+...+1/49-1/50

E=1-1/50

E=49/50

E=1-1/2+1/2-1/3+....+1/49-1/50=1-1/50=49/50

A = 1.2 + 2.3 + ........+49.50

3A = 1.2.(3-0) + 2.3.(4-1)+........+49.50.(51 - 48)

3A = 1.2.3 + 2.3.4 - 1.2.3 +........ + 49.50.51 - 48.49.50

3A = 48.49.50 = 117600

A = 39200 

Ta có :
Gọi A=1.2+2.3+3.4+4.5+...+49.50
 A=1.2+2.3+3.4+4.5+...+49.50
 3.A=3.(1.2+2.3+3.4+4.5+...+49.50)
 3.A=1.2.3+2.3.3+3.3.4+3.4.5+...+3.49.50
 3.A=1.2.(3-0)+2.3.(3-0)+(3-0).3.4+(3-0).4.5+...+(3-0).49.50
 3.A=1.2.3-0+2.3.3-0+3.3.4-0+3.4.5-0+...+3.49.50-0
 3.A=1.2.3-0+2.3.4-1.2.3+5.3.4-2.3.4+...+49.50.51-48.49.50
 3.A=49.50.51
 A= 49.50.51/3
 A= 49.50.17.3/3
 A=49.50.17
 A=41650
Đáp số : A=41650

1x2+2x3+3x4+4x5+...+49x50=A
3.(1.2+2.3+3.4+4.5+...+49.50)=A.3
3.1.2+3.2.3+3.3.4+3.4.5+...+3.49.50+A.3
(3-0).1.2+(4-1).2.3+(5-2).3.4+(6-3).4.5+...+(51-48).49.50=A.3
3.1.2-0.1.2+4.2.3-1.2.3+5.3.4-2.3.4+6.4.5-3.4.5+...+51.49.50-48,49,50=A.3
51.49.50=A.3
51.49.50:3=A
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