Ta có : 
A = 1.2 + 2.3 + 3.4 + ... + 198.199 + 199.200 
= 1.(1 + 1) + 2.(2 + 1) + 3.(3 + 1) + ... + 198(198 + 1) + 199(199 + 1) 
= (1^2 + 1) + (2^2 + 2) + (3^2 + 3) + ... + (198^2 + 198) + (199^2 + 199) 
= (1 + 2 + 3 + 4....+ 198 + 199) + (1^2 + 2^2 + 3^2 + ...+ 198^2 + 199^2) 
* Dễ chứng minh : 
....1 + 2 + 3 +...+ n = n(n + 1)/2 
.... 1^2 + 2^2 +...+ n^2 = [n(n + 1)(2n + 1)]/6 
Suy ra : A = [199.(199 + 1)]/2 + [199.(199 + 1)(2.199 + 1)]/6 = 2666600

Đặt A = 1.2 + 2.3 + 3.4 + .... + 199.200

⇒ 3A = 1.2.3 + 2.3.3 + 3.4.3 + .... + 199.200.3

⇒ 3A = 1.2.3 + 2.3.( 4 - 1 ) + 3.4.( 5 - 2 ) + .... + 199.200.( 201 - 198 )

⇒ 3A = 1.2.3 + 2.3.4 - 1.2.3 + 3.4.5 - 2.3.4 + ... + 199.200.201 - 198.199.200 

⇒ 3A = ( 1.2.3 - 1.2.3 ) + ( 2.3.4 - 2.3.4 ) + .... + ( 198.199.200 - 198.199.200 ) + 199.200.201

⇒ 3A = 199.200.201

⇒ 3A = \(\frac{199.200.201}{3}\)

3A =1.2.3 +2.3.(4-1) +3.4.(5-2) +4.5.(6-3)....+199.200.(201 -198)

    = 1.2.3+2.3.4 -1.2.3 +3.4.5- 2.3.4 + 4.5.6 - 3.4.5 +......+ 199.200.201 -198.199.200

 3A =199.200.201 

A=199.200.67 =254600

S = -5/1.2 + -5/2.3 + -5/3.4 + ... + -5/199.200

S= -5/1 - (-5)/2 +(-5)/2 - (-5)/3+....+ (-5)/199 - (-5)/200

S= -5/1 - (-5)/200

S= -5/1 + 5/200

S= -199/40 

\(S=\frac{-5}{1\cdot2}+\frac{-5}{2\cdot3}+\frac{-5}{3\cdot4}+...+\frac{-5}{199\cdot200}\)

\(S=\frac{-5}{1}-\frac{-5}{2}+\frac{-5}{2}-\frac{-5}{3}+...+\frac{-5}{199}-\frac{-5}{200}\)

\(S=-5+\frac{5}{200}\)

\(S=-\frac{199}{40}\)

Bạn check lại nha, không chắc đâu =.=

Ta có: A=1.2+2.3+...+198.199+199.200 
=>3A=1.2.3+2.3.3+...+198.199.3 
+199.200.3 
=>3A=1.2.3+2.3(4-1)+...+ 
198.199(200-197)+199.200(201-198) 
=>3A=1.2.3+2.3.4-1.2.3+...+198.199.200 
-197.198.199+199.200.201-198.199.200 
=>3A=199.200.201 
=>A=199.200.67

A=39800.67

A=2666600

lấy máy tính mà tính

các bạn troll mình à

\(A=\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+...+\frac{1}{199\cdot200}\)

\(A=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{199}-\frac{1}{200}\)

\(A=1-\frac{1}{200}\)

\(A=\frac{199}{200}\)

\(=1-\frac{1}{200}=\frac{199}{200}\)

\(A=\frac{1}{1.2}+............+\frac{1}{199.200}\)

\(=\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-...........+\frac{1}{199}-\frac{1}{200}\)

\(=1-\frac{1}{200}\)

\(=\frac{199}{200}\)

\(E=1.2+2.3+3.4+..+199.200\)

\(3E=1.2.3+2.3.3+3.4.3+...+199.200.3\)

\(3E=1.2.3+2.3.\left(4-1\right)+3.4.\left(5-2\right)+...+199.200.\left(201-198\right)\)

\(3E=1.2.3+2.3.4+3.4.5+...+199.200.201\)

\(3E=\frac{199.200.201}{3}\)

\(E=2666600\)