Bài 1 :

A = 1² + 2² + 3² +....+n² 

A = 1² + 2.(1+1) + 3.(2 +1) + 4.( 3 +1) +.....+n(n-1 + 1)

A = 1 + 1.2 + 2 + 2.3 + 3 + 3.4 + 4 +.....+ n.(n-1) + n

A = ( 1 + 2 + 3 + 4 +....+n) + ( 1.2 + 2.3 + 3.4 +....+(n-1).n

A = (n+1).{(n-1):n+1)/2 +1/3.[1.2.3 +2.3.3 +.....+(n-1)n.3]

A = (n+1).n/2+1/3.[1.2.3 +2.3.(4-1)+ ...+(n-1).n [(n+1) - (n -2)]

A = (n+1)n/2+1/3.( 1.2.3 + 2.3.4 -1.2.3 +..+ (n-1)n(n+1)- (n-2)(n-1)n)

A =(n+1)n/2 + 1/3.(n-1)n(n+1)

A = n(n+1)[1/2 + 1/3 .(n-1)]

A = n.(n+1) \(\dfrac{3+2n-2}{6}\)

A= n.(n+1)(2n+1)/6

Bài 2 : 

a, (x+1) +(x+2) + (x+3)+...+(x+10) = 5070

    (x+10 +x+1).{( x+10 - x -1): 1 +1):2  = 5070

    (2x + 11)10 : 2 = 5070 

     ( 2x + 11)5 = 5070

      2x+ 11 = 5070:5

         2x = 1014 - 11

        2x =   1003

          x = 1003 :2

          x = 501,5 

        b, 1 + 2 + 3 +...+x = 820

           ( x + 1)[ (x-1):1 +1] : 2 = 820

           (x +1).x = 820 x 2

           (x +1).x = 1640

            (x +1) .x = 40 x 41

                 x = 40 

Thanks trước

helppp me?

2: ĐKXĐ: \(x\ne2\)

\(\dfrac{3x^2+6x+12}{x^3-8}=\dfrac{3\left(x^2+2x+4\right)}{\left(x-2\right)\left(x^2+2x+4\right)}=\dfrac{3}{x-2}\)

Sửa đề: \(B=\dfrac{x-3\sqrt{x}+4}{x-2}-\dfrac{1}{\sqrt{x}-2}\)

Ta có: \(B=\dfrac{x-3\sqrt{x}+4}{x-2}-\dfrac{1}{\sqrt{x}-2}\)

\(=\dfrac{x-3\sqrt{x}+4}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}-\dfrac{\sqrt{x}+2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\)

\(=\dfrac{x-3\sqrt{x}+4-\sqrt{x}-2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\)

\(=\dfrac{x-4\sqrt{x}+2}{x-2}\)

Ta có: \(\dfrac{x^{24}+x^{20}+x^{16}+...+x^4+1}{x^{26}+x^{24}+x^{22}+...+x^2+1}\)

\(=\dfrac{x^{24}+x^{20}+x^{16}+...+x^4+1}{\left(x^{26}+x^{22}+...+x^2\right)+\left(x^{24}+x^{20}+x^{16}+...+x^4+1\right)}\)

\(=\dfrac{x^{24}+x^{20}+x^{16}+...+x^4+1}{x^2\left(x^{24}+x^{20}+...+1\right)+\left(x^{24}+x^{20}+x^{16}+...+x^4+1\right)}\)

\(=\dfrac{x^{24}+x^{20}+x^{16}+...+x^4+1}{\left(x^{24}+x^{20}+x^{16}+...+1\right)\left(x^2+1\right)}\)

\(=\dfrac{1}{x^2+1}\)

 =x24+x20+x16+...+x4+1(x26+x22+...+x2)+(x24+x20+x16+...+x4+1)=x24+x20+x16+...+x4+1(x26+x22+...+x2)+(x24+x20+x16+...+x4+1)

=x24+x20+x16+...+x4+1(x24+x20+x16+...+1)(x2+1)

\(=x^2+2x+1-2x^2+18+x^2-4x+4\\ =-2x+23=-2\cdot12+23=-24+23=-1\)

\(\dfrac{1}{\sqrt{x}-1}+\dfrac{\sqrt{x}}{\sqrt{x}+3}-\dfrac{12}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-1\right)}\left(\text{đ}k\text{x}\text{đ}:x\ge0;x\ne1\right)\\ =\dfrac{\sqrt{x}+3}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+3\right)}+\dfrac{\sqrt{x}\left(\sqrt{x}-1\right)}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-1\right)}-\dfrac{12}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-1\right)}\\ =\dfrac{\sqrt{x}+3+x-\sqrt{x}-12}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-1\right)}\\ =\dfrac{x-9}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-1\right)}\\ =\dfrac{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-1\right)}\\ =\dfrac{\sqrt{x}-3}{\sqrt{x}-1}\)

\(=\dfrac{\sqrt{x}+3+\sqrt{x}\left(\sqrt{x}-1\right)-12}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-1\right)}\)

\(=\dfrac{\sqrt{x}-9+x-\sqrt{x}}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-1\right)}=\dfrac{x-9}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-1\right)}\)

\(=\dfrac{\sqrt{x}-3}{\sqrt{x}-1}\)

`(1)/(\sqrt{x}+1)+(\sqrt{x})/(\sqrt{x}+3)-(12)/((\sqrt{x}+3)(\sqrt{x}-1))`

Điều Kiện `x>=0;x\ne1`

`=(\sqrt{x}+3+\sqrt{x}(\sqrt{x}-1)-12)/((\sqrt{x}+3)(\sqrt{x}-1))`

`=(\sqrt{x}+3+x-\sqrt{x}-12)/((\sqrt{x}+3)(\sqrt{x}-1))`

`=(x-9)/((\sqrt{x}+3)(\sqrt{x}-1))`

`=((\sqrt{x}-3)(\sqrt{x}+3))/((\sqrt{x}+3)(\sqrt{x}-1))`

`=(\sqrt{x}-3)/(\sqrt{x}-1)`

b: \(=\left(x^2+3x+1-3x+1\right)^2=\left(x^2+2\right)^2\)

Bài 2:

a: 2/6x5/3=10/18=5/9

b: 11/9x5/10=55/90=11/18

c: 3/9x6/8=1/3x3/4=1/4

d: 4/9x12/16=48/144=1/3

e: 25/15x6/7=5/3x6/7=30/21=10/7

f: 6/10x15/20=90/200=9/20

Bài 1

4/5 x 6/7= 24/35

2/9 x 1/2= 2/18= 1/9

1/2 x 8/3= 8/6= 4/3

7/9 x 6/5= 42/45= 14/15

8/7 x 5/9= 40/63

10/11 x 22/15= 220/165= 4/3

Bài 2

2/6 x 5/3= 1/3 x 5/3=5/9

11/9 x 5/10= 11/9 x 1/2= 11/18

3/9 x 6/8= 1/3 x 3/4 =3/12= 1/4

4/9 x 12/16= 4/9 x 3/4= 12/36= 1/3

25/15 x 6/7= 5/3 x 6/7= 30/21= 10/7

6/10 x 15/20= 3/5 x 3/4= 9/20

bài 1

4/5 x 6/7 = 24/35

2/9 x 1/2 = 1/9

1/2 x 8/3 = 4/3

7/9 x 6/5 = 14/15

8/7 x 5/9 = 14/15

8/7 x 5/9 = 40/63

10/11 x 22/15 = 4/3