1/1.2+1/2.3+1/3.4+...+1/59.60

=1-1/2+1/2-1/3+1/3-1/4+...+1/59-1/60

=1-1/60

=59/60

vì 1>59/60

=> 1>1/1.2+1/2.3+1/3.4+...+1/59.60

chúc bạn học tốt nha

\(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{59.60}\)

\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{59}-\frac{1}{60}\)

\(=1-\frac{1}{60}=\frac{59}{60}\)

Đặt \(C=\frac{1}{1\times2}+\frac{1}{2\times3}+\frac{1}{3\times4}+...+\frac{1}{59\times60}\)

          \(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+....+\frac{1}{59}-\frac{1}{60}\)

          \(=1-\frac{1}{60}< 1\)

=> C < 1 (đpcm)

1/2x3+1/3x4+1/4x5+.......+1/57x58+1/58x59+1/59x60 

1/2-1/3+1/3-1/4 +1/4 - 1/5 +... + 1/59-1/60

=1/2-1/60

=29/60

\(=\frac{29}{60}\) tích nha !

A= 1x2+2x3+3x4+...+98x99 A x 3= 1x2 x (3-0) +2x3x (4-1)+3x4 x (5-2)+...+98x99x (100-97) = 1x2x3+2x3x4+......98x99x100- (1x2x0+ 2x3x1+....+ 98x99x97) = 98x99x100

A= 1x2+2x3+3x4+...+98x99
A x 3= 1x2 x (3-0) +2x3x (4-1)+3x4 x (5-2)+...+98x99x (100-97)
= 1x2x3+2x3x4+......98x99x100- (1x2x0+ 2x3x1+....+ 98x99x97)
= 98x99x100.

có công thức nè bạn 1²+2²+3³+...+n²= n(n+1)(2n+1):6

1x 2 + 2 x 3 + 3 x 4 + ...+ 99 x 100

Ta có: 

1 x 2 x 3                                           = 1 x 2 x 3

2 x 3 x 3     = 2 x 3 x ( 4 - 1)             = 2 x 3 x 4 -  1 x 2 x 3

3 x 4 x 3     =  3 x 4 x ( 5 - 2)            = 3 x 4 x 5 -   2 x 3 x 4 

........................................................= ........................................

99 x 100 x 3 = 99 x 100 x (101 - 98)  = 99 x 100 x 101 - 99 x 100 x 98

Cộng vế với vế ta có:

1 x 2 x 3 + 2 x 3 x 3 + 3 x 4 x 3 +...+ 99 x 100 x 3 = 99 x100 x 101

(1 x 2 + 2 x 3 + 3 x 4 +...+ 99 x 100) x 3 =  99 x 100 x 101 

1 x 2 + 2 x 3 + 3 x 4 +...+ 99 x 100 = \(\dfrac{99\times100\times101}{3}\)

1 x 2 + 2 x 3 + 3 x 4 + ....+ 99 x 100 = 333300

333300

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 = 1.2 + 2.3 + .... + 999.1000

3A = 1.2.3 + 2.3.(4-1) + .... + 999.1000.(1001 - 998)

3A = 1.2.3 + 2.3.4 - 1.2,3 +..... + 999.1000.1001 - 998.999.1000

3A = 999 . 1000 . 1001

A = 333 x 1000 x 1001 = 333 333 000

3A=1x2x3+2x3x(4-1)+3x4x(5-2)+.............+999x1000x(1001-998)

3A=1x2x3+2x3x4-1x2x3+3x4x5-2x3x4+............+999x1000x1001-998x999x1000

3A=999x1000x1001

A=999x1000x1001:3

A=333333000

=> 3A = 1.2.3 + 2.3.(4 - 1) + 3.4.(5 - 2) + ................ + 999.1000.(1001 - 998)

=> 3A = 1.2.3 + 2.3.4 - 1.2.3 + 3.4.5 - 2.3.4 + ............... + 999.1000.1001 - 998.999.1000

=> 3A = 999.1000.1001

=> A = 999.1000.1001 = 333333000

s=2-1+3-2+...+39-38+40-39

  =40-1

 = 39

thank you

\(S=1.2+2.3+..........+39.40\)

\(3S=1.2.3+2.3.3+...........+39.40.3\)

\(3S=\left(3-0\right).1.2+\left(4-1\right).2.3.+......+\left(41-38\right).39.40.\)

\(3S=1.2.3-0.1.2+2.3.4-1.2.3+......+39.40.41-38.39.40\)

\(3S=0+39.40.41\)

\(3S=63960\)

   \(S=63960:3=21320\)