\(S=\left(\dfrac{1}{1\cdot3}+\dfrac{1}{3\cdot5}+...+\dfrac{1}{97\cdot99}\right)+\left(\dfrac{1}{2\cdot4}+\dfrac{1}{4\cdot6}+...+\dfrac{1}{98\cdot100}\right)-\dfrac{49}{99}\)

\(=\dfrac{1}{2}\left(\dfrac{2}{1\cdot3}+\dfrac{2}{3\cdot5}+...+\dfrac{2}{97\cdot99}\right)+\dfrac{1}{2}\left(\dfrac{2}{2\cdot4}+\dfrac{2}{4\cdot6}+...+\dfrac{2}{98\cdot100}\right)-\dfrac{49}{99}\)

\(=\dfrac{1}{2}\left(1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+...+\dfrac{1}{97}-\dfrac{1}{99}\right)+\dfrac{1}{2}\left(\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{6}+...+\dfrac{1}{98}-\dfrac{1}{100}\right)-\dfrac{49}{99}\)

\(=\dfrac{1}{2}\cdot\dfrac{98}{99}+\dfrac{1}{2}\cdot\dfrac{49}{100}-\dfrac{49}{99}\)

\(=\dfrac{49}{200}\)

B=1.3+2.4+3.5+...+97.99+98.100B=1.3+2.4+3.5+...+97.99+98.100

B=1(2+1)+2(3+1)+....+97(98+1)+98(99+1)B=1(2+1)+2(3+1)+....+97(98+1)+98(99+1)

B=1.2+1+2.3+2+....+97.98+97+98.99+98B=1.2+1+2.3+2+....+97.98+97+98.99+98

B=(1.2+2.3+3.4+....+97.98+98.99)+(1+2+3+...+98)B=(1.2+2.3+3.4+....+97.98+98.99)+(1+2+3+...+98)

B=98.99.1003+98.992B=98.99.1003+98.992

B=323400+4851=328251B=323400+4851=328251 

Số đó=1.3 + 2.4 + 3.5 +....+ 98.100 
= 1(2+1) + 2.(3+1) + 3.(4+1) +...+ 98(99+1)
= 1.2 + 1 + 2.3 + 2 + 3.4 + 3+....+ 98.99 +98
= (1.2 + 2.3 + 3.4+....98.99) + (1+2+3+....+98)
=323400 + 4851=328251

=1-1/3-1/2+1/4+1/3-1/5-1/4+1/6+...+1/97-1/99-1/98+1/100

=1-1/2-1/99-1/98=2327/4851

\(\frac{1}{1.3}-\frac{1}{2.4}+\frac{1}{3.5}-\frac{1}{4.6}+...+\frac{1}{97.99}-\frac{1}{98.100}\)

\(=1-\frac{1}{3}-\frac{1}{2}+\frac{1}{4}+\frac{1}{3}-\frac{1}{5}-\frac{1}{4}+\frac{1}{6}+...+\frac{1}{97}-\frac{1}{99}-\frac{1}{98}+\frac{1}{100}\)

\(=1-\frac{1}{2}-\frac{1}{99}-\frac{1}{98}\)

\(=\frac{2327}{4851}\)

Đặt A=1/1.3 - 1/2.4 +1/3.5 -1/4.6 +.....+1/97.99 -1/98.100

     4A= 4/1.3 -4/2.4 +4/3.5 -4/4.6 +.....+4/97.99 -4/98.100

          =(4/1.3 +4/3.5 +...+4/97.99) - (4/2.4 +4/4.6 +...+4/98.100)

          =(1/1 -1/3+1/3-1/5+...+1/97-1/99)-(1/2 -1/4 -....1/98-1/100)

         =(1/1-1/99)-(1/2-1/100)

         4A=98/99 - 99/100

         A= (98/99-99/100) :4

ngắn gọn nhất nè
1/1.3-1/2.4+1/3.5-1/4.6+...+1/97.99-1/98.100 
= 1- 1/3 - 1/2 + 1/4 + 1/3 - 1/5 - 1/4 + 1/6 + ... + 1/97 - 1/99 - 1/98 + 1/100
= 1 - 1/2 - 1/99 - 1/98
= 2327/4851

=>S<1/1.2+1/2.3+1/3.4+...+1/98.99+1/99.100=1-1/2+1/2-1/3+1/3-1/4+...+1/99-1/100

                                                                  =1-(1/2-1/2)-(1/3-1/3)-...-(1/99-1/99)-1/100

                                                                  =1-1/100 <1

=>S<1

Vậy S<1

1/1.3 - 1/2.4 + 1/3.5 - 1/4.6 + ...+ 1/97.99 - 1/98.100 + 3lxl = 1 

tách ra trc đầu tiên tính phần : 1/1.3 - 1/2.4 + 1/3.5 - 1/4.6 + ...+ 1/97.99 - 1/98.100

tách số lẻ và số chẵn ra 

(1/1.3+1/3.5+...+1/57.97+1/97.99)-(1/2.4+1/4.6+...1/98.100)

tính từng vế  vế đầu kết qu3 vế lẻ là : 49/99 

kết quả vế chẵn là 49/200

thì bài đó sẻ thành : 49/99+49/200+3lxl=1 

còn lại tự tinh nha