Lời giải :

Đặt S=1.2+2.3+3.4+4.5+…+99.100+100.101

3S=1.2.3+2.3.3+3.4.3+4.5.3+…+99.100.3+100.101.3

=1.2(3−0)+2.3(4−1)+3.4(5−2)+4.5(6−3)+…+99.100(101−98)+100.101(102−99)

=0.1.2-1.2.3+1.2.3-2.3.4+...+99.100.101-100.101.102

=100.101.102

S=100.101.34=343400

1.Tính 

a) Ta có: 

  A=(1-1/2²).(1-1/3²)...(1-1/100²)

=>A=3/2².8/3².....9999/100²

=>A=(1.3/2.2).(2.4/3.3).....(99.101/100.100)

=>A=(1.2.3.....99/2.3.4.....100).(3.4.5.....101/2.3.4.....100)

=>A=1/100.101/2

=>A=101/200

b) Ta có: 

  B=-1/1.2-1/2.3-1/3.4-...-1/100.101

=>B=-(1/1.2+1/2.3+1/3.4+...+1/100.101)

=>B=-(1-1/2+1/2-1/3+1/3-1/4+...+1/100-1/101)

=>B=-(1-1/101)

=>B=-100/101

 c) Ta có:

 C=1.2+2.3+3.4+...+100.101

       =>3C=1.2.3+2.3.3+3.4.3+...+100.101.3

       =>3C=1.2.3+2.3.(4-1)+3.4.(5-2)+...+100.101.(102-99)

       =>3C=1.2.3-1.2.3+2.3.4-2.3.4+3.4.5-3.4.5+...+100.101.102

       =>3C=100.101.102

       =>3C=1030200

       =>C=343400

Chúc bạn hok tốt nhé >:)!!!!!

A = 1.2 + 2.3 + 3.4 + ...... + 100.101

3A = 1.2.3 + 2.3.3 + 3.4.3 + ...... + 100.101.3

3A = 1.2.3 + 2.3.(4 - 1) + 3.4.(5 - 2) + ..... + 100.101.(102 - 99)

3A = 1.2.3 + 2.3.4 - 1.2.3 + 3.4.5 - 2.3.4 + ...... + 100.101.102 - 99.100.101

3A = 100.101.102

A = 100.101.34

A = 343400

\(3A=1.2.3+2.3.\left(4-1\right)+...+100.101.\left(102-99\right)\)

\(3A=1.2.3+2.3.4-1.2.3+.......+100.101.102-99.100.101\)

\(3A=100.101.102\)

\(A=\frac{100.101.102}{3}\)

\(A=343400\)

3=1.2.3+2.3(4-1)+...+100.101(102-99)

3=1.2.3+2.3.4-1.2.3+.....+100.101.102-99.100.101

3=100.101.101

=100.101.102/3

=343400

mn ủng hộ ^--^

đáp án:

bằng 343400

học tốt

Đặt A = 1.2 + 2.3 + 3.4 + 4.5 + ... + 99.100 + 100.101

3A = 1.2.3 + 2.3.3 + 3.4.3 + 4.5.3 + ... + 99.100.3 + 100.101.3

3A = 1.2.3 + 2.3.(4 - 1) + 3.4.(5 - 2) + ... + 99.100.(101 - 98) + 100.101.(102 - 99)

3A = 1.2.3 + 2.3.4 - 1.2.3 + 3.4.5 - 2.3.4 + ... + 99.100.101 - 98.99.100 + 100.101.102 - 99.100.101

3A = 100.101.102

3A = 1030200

A = 343400

Đặt A=1.2+2.3+3.4+...+99.100+100.101

     3A=1.2.3+2.3.3+3.4.3+...+99.100.3+100.101.3

         =1.2.3+2.3.(4-1)+3.4.(5-2)+...+99.100.(101-98)+100.101.(102-99)

         =1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+...+99.100.101-98.99.100+100.101.102-99.100.101

        =100.101.102

\(\Rightarrow A=\frac{100.101.102}{3}\)

Gọi biểu thức này là S , ta có 

S =1.2 + 2.3 + 3.4 + 4.5 + ...+ 100.101

3S= 1. 2 .3 + 2. 3 .3 + 3 . 4 .3 + 4 .5 .3 + ...........+ 100 .101 .3

3S= 1.2 (3 - 0) + 2 . 3 .(4 - 1) + 3 . 4. (5 - 2 ) +.......+ 100 . 101 . (102 - 99)

3S = 1 . 2 . 3 - 0 . 1 .2  + 2 . 3 . 4 - 1 . 2 .3 + ................+ 100 . 101 .102 - 99  100 . 101 

S   =   \(\frac{100.101.102}{3}=\frac{100.101.34}{1}\)

S   = 343400

Ta có: 
A-B =1.2+2.3+3.4+...+100.101- 
(1^2+2^2+3^2+4^2+...+100^2) 
= 1.2+2.3+3.4+...+100.101- 

Đặt B = 1.2+2.3 +.......+99.100+100.101

3B= 1.2.3+2.3.4+3.4.3 +......+ 99.100.3 + 100.101.3

3B= 1.2. (3 - 0) + 2.3.(4 - 1) +3.4. (5 - 2)....... . 99.100. (101 - 98) . 100.101.(102 - 99)

3B = (1.2.3 + 2.3.4 + 3.4.5 +...... + 99.100.101 + 100.101.102) - (0.1.2 + 1.2.3 + 2.3.4 +.......+ 98.99.100.99.100.101)

3B = 100.101.102 - 0.1.2

3B = 1030200 - 0

3B= 1030200

B =  1030200 : 3

B =  343400

đặt A = 1.2+2.3+3.4+.....+100.101

=> 3A = 1.2.3+2.3.3+.......+100.101.3

=> 3A = 1.2.3+2.3.(4-1)+......+100.101.(102-99)

=> 3A = 1.2.3 + 2.3.4-1.2.3 + ........+ 100.101.102 - 99.100.101

=> 3A = 100.101.102

=> A = 100.101.102 : 3

=> A = 343400

ĐẶT S = 1.2 + 2.3 +3.4 +........ + 100.101 

3S = 1 . 2 .3 + 2 . 3 . 3+ 3.4.3 + ....... + 99 . 100 . 3

3S= 1 . 2 . 3 + 2. 3 . ( 4-1 ) + .........  + 99 . 100 . (101 -98 )

3S= 1 . 2 . 3 + 2 . 3 . 4 - 1 . 2 . 3+ 3 . 4 . 5 - 2 . 3 . 4 + ......   + 99 . 100 . 101 - 98 . 99 . 100

3S = 99 . 100 . 101

3S = 3 .33 . 100 . 101

S= 33 . 100 . 101 = 333300

Đặt A = 1.2+2.3+3.4+...+100.101

3A = 1.2.3+2.3.3+3.4.3+...+100.101.3

3A= 1.2.(3-0)+2.3.(4-1)+3.4.(5-2)+...+100.101.(102-99)

3A= 1.2.3-0.1.2+2.3.4-1.2.3+...+100.101.102-99.100.101

3A= (1.2.3-1.2.3)+(2.3.4-2.3.4)+...+(99.100.101-99.100.101)+(100.101.102-0.1.2)

3A= 0+0+...+0+(100.101.102-0.1.2)

3A= 100.101.102-0.1.2

3A= 100.101.102

A= (100.101.102):3

A= 1030200:3

A= 343400

Tớ ngắn gọn hơn nè

B=1.2+2.3+3.4+...+100.101

=>3B=1.2.3+2.3.3+3.4.4+....+100.101.3

=1.2.3-0.1.2+2.3.4-1.2.3+3.4.5-2.3.4+....+100.101.102-99.100.101

=-0.1.2-100.101.102

=1030200

=>B=1030200:3=343400

B = 1.2+2.3 +.......+99.100+100.101

3B= 1.2.3+2.3.4+3.4.3 +......+ 99.100.3 + 100.101.3

3B= 1.2. (3 - 0) + 2.3.(4 - 1) +3.4. (5 - 2)....... . 99.100. (101 - 98) . 100.101.(102 - 99)

3B = (1.2.3 + 2.3.4 + 3.4.5 +...... + 99.100.101 + 100.101.102) - (0.1.2 + 1.2.3 + 2.3.4 +.......+ 98.99.100.99.100.101)

3B = 100.101.102 - 0.1.2

3B = 1030200 - 0

3B= 1030200

B =  1030200 : 3

B =  343400

3A = 1.2.3+2.3(4-1)+3.4.(5-2)+.+99.100.(101-98)

 3A = 1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+.+99.100.101-98.99.100

 3A = 99.100.101