(1+1) + (1+2) + (1+2+3) + (1+2+3+4) + ... + (1+2+3+4+...+99)

Ta có thể nhận thấy rằng mỗi mục trong dãy có thể được biểu diễn dưới dạng tổng của các số từ 1 đến n, trong đó n tăng dần từ 1 đến 99. Vậy ta có thể viết lại dãy số ban đầu như sau:

(1) + (1+2) + (1+2+3) + (1+2+3+4) + ... + (1+2+3+4+...+99)

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

= 1*(1) + 2*(1+2) + 3*(1+2+3) + 4*(1+2+3+4) + ... + 99*(1+2+3+4+...+99)

= 1*(1) + 2*(1+2) + 3*(1+2+3) + 4*(1+2+3+4) + ... + 99*(1+2+3+4+...+99)

= 11 + 23 + 36 + 410 + ... + 99*(1+2+3+4+...+99)

= 11 + 2(1+2) + 3*(1+2+3) + 4*(1+2+3+4) + ... + 99*(1+2+3+4+...+99)

= 11 + 21 + 22 + 31 + 32 + 33 + 41 + 42 + 43 + 44 + ... + 99*(1+2+3+4+...+99)

= 1^2 + 2^2 + 3^2 + 4^2 + ... + 99^2

Vậy, tổng của dãy số ban đầu là tổng bình phương của các số từ 1 đến 99.

1/2* x+2/3=9/2

1/2 * x = 9/2 - 2/3 

1/2 * x= 23/6

x= 23/6 : 1/2

x= 23/6 x 2= 23/3

___

1/2*x-1/3=2/3

1/2*x = 2/3 + 1/3

1/2 * x= 1

x= 1: 1/2 

x= 2

____

1/4+3/4:x=3

3/4 : x = 3 - 1/4

3/4 : x= 11/4

x= 11/4 : 3/4

x= 11/3

\(\dfrac{1}{2}\)\(\times\)\(x\) + \(\dfrac{2}{3}\) = \(\dfrac{9}{2}\)

\(\dfrac{1}{2}\)\(\times\)\(x\)        = \(\dfrac{9}{2}\) - \(\dfrac{2}{3}\)

\(\dfrac{1}{2}\)\(\times\)\(x\)       = \(\dfrac{23}{6}\)

      \(x\)       = \(\dfrac{23}{6}\):\(\dfrac{1}{2}\)

      \(x\)      = \(\dfrac{23}{3}\) 

\(\dfrac{1}{2}\)\(\times\)\(x\) - \(\dfrac{1}{3}\) = \(\dfrac{2}{3}\) 

\(\dfrac{1}{2}\)\(\times\)\(x\)       = \(\dfrac{2}{3}\) + \(\dfrac{1}{3}\)

\(\dfrac{1}{2}\times\)\(x\)      =  1

     \(x\)       = 1 : \(\dfrac{1}{2}\)

   \(x\)         = 2

\(\dfrac{1}{4}\) + \(\dfrac{3}{4}\): \(x\) = 3

          \(\dfrac{3}{4}\): \(x\) = 3 - \(\dfrac{1}{4}\) 

          \(\dfrac{3}{4}\):\(x\) = \(\dfrac{11}{4}\)

              \(x\) = \(\dfrac{3}{4}\): \(\dfrac{11}{4}\)

             \(x\) = \(\dfrac{3}{11}\)

` @ L I N H `

1/2* x+2/3=9/2

1/2 * x = 9/2 - 2/3 

1/2 * x= 23/6

x= 23/6 : 1/2

x= 23/6 x 2= 23/3

___

1/2*x-1/3=2/3

1/2*x = 2/3 + 1/3

1/2 * x= 1

x= 1: 1/2 

x= 2

____

1/4+3/4:x=3

3/4 : x = 3 - 1/4

3/4 : x= 11/4

x= 11/4 : 3/4

x= 11/3

\(B=1+\frac{1}{2}\left(1+2\right)+\frac{1}{3}\left(1+2+3\right)+...+\frac{1}{x}\left(1+2+...+x\right)\)

\(=1+\frac{1}{2}\cdot\frac{2\cdot3}{2}+\frac{1}{3}\cdot\frac{3\cdot4}{2}+\frac{1}{4}+\frac{4\cdot5}{2}+...+\frac{1}{x}\cdot\frac{x\left(x+1\right)}{2}\)

\(=1+\frac{3}{2}+\frac{4}{2}+\frac{5}{2}+...+\frac{x+1}{2}\)

\(=\frac{1}{2}\left(2+3+4+...+x+1\right)\)

\(=\frac{1}{2}\cdot\frac{\left(x+1+2\right)\left(x+1-2+1\right)}{2}\)

\(=\frac{1}{2}\cdot\frac{x\left(x+3\right)}{2}=\frac{x\left(x+3\right)}{4}\).

7/4

62/72

1/3

5/2

5/2-3/4=7/4

5/2 - 3/4 =20/8-6/8=14/8

1/9+6/8=8/72+54/72=62/72

2/3-1/3=1/3

4/2+1/2=5/2

\(\left(1-\dfrac{1}{2^2}\right)\left(1-\dfrac{1}{3^2}\right)\left(1-\dfrac{1}{4^2}\right)...\left(1-\dfrac{1}{10^2}\right)\)

= \(\dfrac{2^2-1}{2^2}.\dfrac{3^2-1}{3^2}.\dfrac{4^2-1}{4^2}...\dfrac{10^2-1}{10^2}\)

= \(\dfrac{1.3}{2^2}.\dfrac{2.4}{3^2}.\dfrac{3.5}{4^2}...\dfrac{9.11}{10^2}\)

= \(\dfrac{\left(1.2.3...9\right).\left(3.4.5...11\right)}{\left(2.3.4...10\right)\left(2.3.4...10\right)}\)

= \(\dfrac{1.11}{10.10}=\dfrac{11}{100}\)

\(C=\left(1-\dfrac{1}{1+2}\right)\left(1-\dfrac{1}{1+2+3}\right)\left(1-\dfrac{1}{1+2+3+4}\right)...\left(1-\dfrac{1}{1+2+3+...+2016}\right)\)

\(=\left(1-\dfrac{1}{3}\right)\left(1-\dfrac{1}{6}\right)...\left(1-\dfrac{1}{\dfrac{\left(2016+1\right).2016}{2}}\right)\)

\(=\left(1-\dfrac{1}{3}\right)\left(1-\dfrac{1}{6}\right)...\left(1-\dfrac{1}{2033136}\right)\)

\(=\dfrac{2}{3}.\dfrac{5}{6}...\dfrac{2033135}{2033136}\)

\(=\dfrac{4}{6}.\dfrac{10}{12}...\dfrac{4066270}{4066272}\)

\(=\left(\dfrac{1}{2}.\dfrac{2}{3}...\dfrac{2015}{2016}\right).\left(\dfrac{4}{3}.\dfrac{5}{4}...\dfrac{2018}{2017}\right)\)

\(=\dfrac{1}{2016}.\dfrac{2018}{3}=\dfrac{1009}{3024}\)

\(C=5\frac{9}{10}:\frac{3}{2}-\left(2\frac{1}{3}.4\frac{1}{2}-2.2\frac{1}{3}\right):\frac{7}{4}\)

\(=\frac{59}{10}:\frac{3}{2}-\left(\frac{7}{3}.\frac{9}{2}-2.\frac{7}{3}\right):\frac{7}{4}\)

\(=\frac{59}{15}-\left[\frac{7}{3}\left(\frac{9}{2}-2\right)\right]:\frac{7}{4}\)

\(=\frac{59}{15}-\frac{35}{6}:\frac{7}{4}\)

\(=\frac{59}{15}-\frac{10}{3}\)

\(=\frac{9}{15}=\frac{3}{5}\)

\(\cdot62,87+35,14+4,13+8,35+4,86+5,65\)

\(=\left(62,87+4,13\right)+\left(35,14+4,86\right)+\left(8,35+5,65\right)\)

\(=67+40+14\)

\(=121\)

giúp mình đi mà mọi người

a.1/6+1/3=1/2

b.3/4-1/2=`1/4

1/2

1/4

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