\(C=1.4+2\left(4+1\right)+3\left(4+2\right)+...+99\left(4+98\right)\)

\(\Leftrightarrow C=1.4+2.4+1.2+3.4+2.3+...+99.4+98.99\)

\(C=\left(1.4+2.4+3.4+....+99.4\right)+\left(1.2+2.3+3.4+..98.99\right)\)

\(C=4\left(1+2+3+...+99\right)+\dfrac{1.2.3+2.3.3+3.4.3+...+98.99.3}{3}\)

\(C=\dfrac{4.\left(1+99\right).99}{2}+\dfrac{1.2.3+2.3\left(4-1\right)+...+98.99\left(100-97\right)}{3}\)

\(C=\dfrac{4.\left(1+99\right).99}{2}+\dfrac{1.2.3+2.3.4-1.2.3...+98.99.100-97.98.99}{3}\)

\(C=\dfrac{4.\left(1+99\right).99}{2}+\dfrac{98.99.100}{3}\)

\(C=19800+107800=127600\)

TK

S=1.4+2.5+3.6+4.7+....+n.(n+3) S = 1. ( 2 + 2 ) + 2. ( 3 + 2 ) + 3. ( 4 + 2 ) + . . . + n . [ ( n + 1 ) + 2 ] S = 1.2 + 2.3 + 3.4 + . . . . + n . ( n + 1 ) + ( 1.2 + 2.2 + 3.2 + . . . . + n .2 ) Đặt A = 1.2 + 2.3 + 3.4 + . . . . + n . ( n + 1 ) 3 A = 1.2.3 + 2.3. ( 4 − 1 ) + . . . . + n . ( n + 1 ) . [ ( n + 2 ) − ( n − 1 ) 3 A = 1.2.3 + 2.3.4 − 1.2.3 + . . . . + n . ( n + 1 ) . ( n + 2 ) − ( n − 1 ) . n . ( n + 1 ) 3 A = n . ( n + 1 ) . ( n + 2 ) A = [ n . ( n + 1 ) . ( n + 2 ) ] : 3 S = [ n . ( n + 1 ) . ( n + 2 ) ] : 3 + 2. ( 1 + 2 + 3 + . . . + n ) S = [ n . ( n + 1 ) . ( n + 2 ) ] : 3 + 2. n . ( n + 1 ) : 2 S = n . ( n + 1 ) . ( n + 2 ) : 3 + n . ( n + 1 ) S = n . ( n + 1 ) . [ ( n + 2 ) : 3 + 1 )

D = 1^2 + 2^2 + 3^2 + ... + n^2 
   = 1.( 2 - 1 ) + 2.( 3-1 ) + 3.( 4-1 ) + .... + n.[ ( n+ 1) - 1 ]
   = 1.2 - 1 + 2.3 - 2 + 3.4 - 3 + .... + n.( n+1 ) - n

   = [ 1.2 + 2.3 + 3.4 + ..... + n.( n + 1 ) ] - ( 1 + 2 + 3 + .... + n ) 
   = { [ n.( n+1 ).( n+2 )] /3 } - { [ n.( n+1)] /2 } 
   = { n(n+1)(2n+1) }/ 6 
Vậy......... 

TK

TK

D=12+22+32+...+n2

D =1.4+2.5+3.6+.......+99.102

D = 1. (2+2) +2.(2+3) +3.(2+4)+...+99.(100+2)

D = 1.2+1.2+2.2+2.3+2.3+3.4+...+2.99+99.100

D = (1.2+2.3+3.4+...+99.100) +2.(1+2+3+4+...+99)

*Gọi A= 1.2+2.3+3.4+...+99.100

3A = 3.(1.2+2.3+3.4+...+99.100)

3A = 1.2.3+2.3.3+...+99.100.3

3A = 1.2.3 +2 .3.(4-1)+...+99.100.(101-98)

3A = 1.2.3+2.3.4-1.2.3+...+ 99.100.101-98.99.100

3A = 99.100.101

3A = 3.33.100.101

A   =  33.100.101

A   = 333300

* Gọi B = 2. (1+2+3+4+...+99) 

                  \__có 99 số hạng ___/          

 B=  2.[(1+99).99:2]

 B = 2 .4950

 B = 9900

A+B = 333300+9900 =343200

Vậy D =343200

A = 1(2+2)+2(3+2)+3(4+2)+.+99(100+2)

A = 1.2+1.2+2.3+2.2+3.4+3.2+.+99.100+99.2

A = (1.2+2.3+3.4+.+99.100)+2(1+2+3+.+99)

Phần còn lại bạn tự động não đi nha.

C= (4/1.3).(9/2.4).(16/3.5)...........(10000/99.101)
C=(4.9.16...........10000)/(1.3).(2.4)......(99.101)
C=(2^2.3^2.4^2..........100^2)/(1.2.3.4......99).(3.4.5.......101)
C=(2.3.4........100).(2.3.4......100)/(1.2.3.....99).(3.4.5....101)
Sau khi triệt tiêu ở tử và mẫu ta được:
C=(2.100)/101
C=200/101
 

1.4+2.5+3.6+...+99.102=1(2+2)+2(3+2)+3(4+2)+...99(100+2)

=1.2+1.2+2.3+2.2+...+99.100+99.2

=(1.2+2.3+...+99.100)+2(1+2+...+99)

A=1.2+2.3+3.4+...+99.100(cho A la ten bieu thuc nay)

3A=1.2(3-0)+2.3(4-1)+...+99.100(101-98)

=(1.2.3+2.3.4+...+99.100.101)-(1.2.3+2.3.4+3.4.5+...+98.99.100

=99.100.101=>A=\(\frac{99.100.101}{3}\)=33330

2.(1+2+...99)

=2(100.99:2)=2.4950=9900

33330+9900=343200

vay...

ngu

Đặt \(A=1.4+2.5+3.6+...+100.103\)

\(=1\left(2.2\right)+2\left(3+2\right)+3\left(4+2\right)+...+100\left(101+2\right)\)

\(=1.2+2.3+3.4+...+100.101+\left(1.2+2.2+3.2+...+100.2\right)\)

\(=1.2+2.3+3.4+...+100.101+2\left(1+2+3+...+100\right)\)

\(=1.2+2.3+3.4+...+100.101+2.100\left(100+1\right):2\)

\(=1.2+2.3+3.4+...+100.101+10100\)

Đặt \(B=1.2+2.3+3.4+...+100.101\)

\(\Rightarrow3B=1.2.3+2.3.3+3.4.3+100.101.3\)

\(\Rightarrow3B=1.2.3+2.3\left(4-1\right)+3.4\left(5-2\right)+...+100.101\left(102-99\right)\)

\(\Rightarrow3B=1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+...+100.101.102-99.100.101\)

\(\Rightarrow3B=100.101.102\)

\(\Rightarrow B=343400\)

Khi đó \(A=343400=10100=333300\)

Đặt A = 1.4 + 2.5 + 3.6 + 4.7 + ... + 100.103

3A = 3.(1.2 + 2.3 + 3.4 + ... + 100.101] + 3.(2 + 4 + 6 + ... + 200)

     = 1.2.3 + 2.3.3 + 3.4.3 + ... + 100.101.3 + 3.(2 + 4 + 6 + ... + 200)

\(\Rightarrow\) A  =  100.101.105:3 = 353500

3 . 6 = 3 . 4 + 2 . 3 rùi đấy bạn, bn xét từng tích rùi sẽ thấy thôi.

Sorano Yuuki !!! Mình hiểu rồi . Thì ra người ta tách sai =.= Cảm ơn nhé .

Đáng nhẽ là . Ta thấy 1.4=1.(2+2)

2.5 = 2.(2 + 3)
3.6 = 3.(2 + 4)
4.7 = 4.(2 + 5)
……

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