Lời giải:

Gọi vế trái là $A$

$2A=\frac{2}{2^2}+\frac{2}{4^2}+\frac{2}{6^2}+...+\frac{2}{2022^2}$

Xét số hạng tổng quát:

$\frac{2}{n^2}$. Ta sẽ cm $\frac{2}{n^2}< \frac{1}{(n-1)n}+\frac{1}{n(n+1)}(*)$

$\Leftrightarrow \frac{2}{n^2}< \frac{n+1+n-1}{n(n-1)(n+1)}$

$\Leftrightarrow \frac{2}{n^2}< \frac{2}{(n-1)(n+1)}$

$\Leftrightarrow \frac{2}{n^2}< \frac{2}{n^2-1}$ (luôn đúng)

Thay $n=2,4,...., 2022$ vào $(*)$ ta có:

$\frac{2}{2^2}< \frac{1}{1.2}+\frac{1}{2.3}$

$\frac{2}{4^2}< \frac{1}{3.4}+\frac{1}{4.5}$

.......

Suy ra: $2A< \frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+....+\frac{1}{2022.2023}$

$2A< 1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+....+\frac{1}{2022}-\frac{1}{2023}$

$2A< 1-\frac{1}{2023}< 1$

$\Rightarrow A< \frac{1}{2}$

Đặt B=122+132+...+182B=122+132+...+182A=11⋅2+12⋅3+...+17⋅8A=11⋅2+12⋅3+...+17⋅8

=1−18<1(2)=1−18<1(2)

Từ  ta có: 

Giải:

a) Ta có:

1/2²=1/2.2 < 1/1.2

1/3²=1/3.3 < 1/2.3

1/4²=1/4.4 < 1/3.4

1/5²=1/5.5 < 1/4.5

1/6²=1/6.6 < 1/5.6

1/7²=1/7.7 < 1/6.7

1/8²=1/8.8 <1/7.8

⇒B<1/1.2+1/2.3+1/3.4+1/4.5+1/5.6+1/6.7+1/7.8

   B<1/1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+1/6-1/7+1/7-1/8

   B<1/1-1/8

   B<7/8

mà 7/8<1

⇒B<7/8<1

⇒B<1

b)S=3/1.4+3/4.7+3/7.10+...+3/40.43+3/43.46

   S=1/1-1/4+1/4-1/7+1/7-1/10+...+1/40-1/43+1/43-1/46

   S=1/1-1/46

   S=45/46

Vì 45/46<1 nên S<1

Vậy S<1

Chúc bạn học tốt!

a)\(\dfrac{1}{2^2}<\dfrac{1}{1.2}\)

\(\dfrac{1}{3^3}<\dfrac{1}{2.3}\)

\(...\)

\(\dfrac{1}{8^2}<\dfrac{1}{7.8}\)

Vậy ta có biểu thức:

\(B=\dfrac{1}{2^2}+\dfrac{1}{3^2}+...+\dfrac{1}{8^2}<\dfrac{1}{1.2}+\dfrac{1}{2.3}+...+\dfrac{1}{7.8}\)

\(B= 1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{7}-\dfrac{1}{8}\)

\(B<1-\dfrac{1}{8}=\dfrac{7}{8}<1\)

Vậy B < 1 (đpcm)

a)[98+(-100)+102].(-2)

=(98-100+102).(-2)

=98-100+102.(-2)

=98-100+(-204)

=-2+(-204)

=-206

b)245+[37+(-245)-|-17|]

=245+(37-245-17)

=245+37-245-17

=245-245+37-17

=0+37-17

=37-17

=20

c)|5|-(-5)-|-7|.3

=5+5-7.3

=5+5-21

=10-21

=11

d)-(2-5)-13

=-2+5-13

=3-13

=-10

e)334+(-20)+(2007-334)

=334-20+2007-334

=334-20-(-2007)-334

=334-334+2007-20

=0+2007-20

=2007-20

=1987

f)-1+3-5+7-9+...+103

=-1+3+(-5)+7+(-9)+...+103
=-2+(-2)+(-2)+...+-2(Có tất cả 51 số -2)

=-2.51

=-102

Xog r` đó

a, x=34

b, x=38

c, x=55

d, x=36

minh nhanh nhat

a , x + 18 = 52 

     x         = 52 - 18 

      x        =     34 

b,   x + 24 = 62 

      x       =   62 - 24 

     x         =     38 

c,  27 + x  = 82 

            x =  82 - 27 

             x =    55

d,  x + 7    = 42 

     x          = 42 - 7 

    x          =      35 

chúc bạn học tốt !

a)x+18=52

     x=52-18

     x=34

b)x+24=62

     x=62-24

     x=38

c)27+x=82

        x=82-27

        x=55

d)x+7=42

       x=42-7

       x=35

t nik nhe