sao nhiều vậy!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

F = 1- 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + ... + 1/99 - 1/100

= 1 - 1/100

= 99/100

a) \(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+....+\frac{1}{2006.2007}\)

\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2006}-\frac{1}{2007}\)

\(=1-\frac{1}{2007}\)

\(=\frac{2006}{2007}\)

CẦN GẤP NHÉ MỌI NGƯỜI

+Câu a:

A = 1/1.2 + 1/2.3 + ...+ 1/5.6 + 1

A = 1/1 - 1/2 + 1/2 - 1/3 + ...+ 1/5 - 1/6 + 1

A = 1/1 - 1/6 + 1

A = 6/6 - 1/6 + 6/6

A = 5/6 + 6/6

A = 11/6

Câu b:

B = 1/1.2.3 + 1/2.3.4 + 1/3.4.5 + ...+ 1/98.99.100

B = 1/2. (2/1.2.3 + 2/2.3.4 + ...+ 2/98.99.100)

B = 1/2.(1/1.2 - 1/2.3 + 1/2.3 - 1/3.4 + ...+ 1/98.99 - 1/99.100)

B = 1/2.(1/2 - 1/9900)

B = 1/2.4949/9900

B = 4949/19800

Câu a:

1/1.2 + 1/2.3 + 1/3.4 + 1/4.5 + 1/5.6 + 1

= 1/1 - 1/2 + 1/2 - 1/3 + 1/4 - 1/5 + 1/5 - 1/6 + 1

= 1/1 - 1/6 + 1

= 6/6 - 1/6 + 6/6

= 5/6 + 1

= 11/6

Câu b:

Câu b:

B = 1/1.2.3 + 1/2.3.4 + 1/3.4.5 + ...+ 1/98.99.100

B = 1/2. (2/1.2.3 + 2/2.3.4 + ...+ 2/98.99.100)

B = 1/2.(1/1.2 - 1/2.3 + 1/2.3 - 1/3.4 + ...+ 1/98.99 - 1/99.100)

B = 1/2.(1/2 - 1/9900)

B = 1/2.4949/9900

B = 4949/19800

\(A=\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{18.19.20}=\frac{1}{2}\left(\frac{2}{1.2.3}+\frac{2}{2.3.4}+...+\frac{2}{18.19.20}\right)\)

\(=\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{18.19}-\frac{1}{19.20}\right)=\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{19.20}\right)\)

\(=\frac{1}{2}\left(\frac{1}{2}-\frac{1}{380}\right)=\frac{1}{4}-\frac{1}{760}< \frac{1}{4}\)(ĐPCM)

Câu a:

A = 1.2 + 2.3 + 3.4 + ...+ 99.100

3A = 1.2.3 + 2.3.3 +..+ 99.100.3

1.2.3 = 1.2.3

2.3.3 = 2.3(4 - 1) = 2.3.4 - 1.2.3

3.4.4 = 3.4(5 - 2) = 3.4.5 - 2.3.4

.............................................................

99.100.3 = 99.100.(101 - 98)=99.100.101-98.99.100

Cộng vế với vế ta có:

3A = 99.100.101

A = 99.100.101 : 3

A = 333300

Câu b:

1.3 + 3.5 + 5.7 + ..+ 97.99

6.A = 1.3.6 + 3.5.6 + 5.7.6 + ...+ 97.99.6

1.3.6 = 1.3.(5 + 1) = 1.3.5 + 1.3.1

3.5.6 = 3.5.(7 - 1) = 3.5.7 - 1.3.5

5.7.6 = 5.7.(9 - 3) = 5.7.9 - 3.5.7

.......................................................

97.99.6 = 97.99.(101 - 95) = 97.99.101-95.97.99

Cộng vế với vế ta có:

6A = 1.3.1 + 97.99.101

A = (1.3.1 + 97.99.101) : 6

A = (3 + 9603.101) : 6

A = (3+ 969903) : 6

A = 969906 : 6

A = 161651

H = \(\frac{1}{1.2}-\frac{1}{1.2.3}+\frac{1}{2.3}-\frac{1}{2.3.4}+\frac{1}{3.4}-\frac{1}{3.4.5}+...+\frac{1}{99.100}-\frac{1}{99.100.101}\)

   \(=\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+....+\frac{1}{99.100}\right)-\left(\frac{1}{1.2.3}+\frac{1}{2.3.4}+....+\frac{1}{99.100.101}\right)\)

Đặt G = \(\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+....+\frac{1}{99.100}\right)\)

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

          = \(1-\frac{1}{100}\)

           = \(\frac{99}{100}\)

Đặt K = \(\left(\frac{1}{1.2.3}+\frac{1}{2.3.4}+....+\frac{1}{99.100.101}\right)\)

=>2K = \(\left(\frac{2}{1.2.3}+\frac{2}{2.3.4}+....+\frac{2}{99.100.101}\right)\)

          = \(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{99.100}-\frac{1}{100.101}\)

          = \(\frac{1}{1.2}-\frac{1}{100.101}\)

          = \(\frac{1}{2}-\frac{1}{10100}\)

          = \(\frac{5049}{10100}\)

=> K =\(\frac{5049}{10100}:2=\frac{5049}{10100}.\frac{1}{2}=\frac{5049}{20200}\)

Thay G,K vào H ta có :

H = \(\frac{99}{100}-\frac{5049}{20200}\)

Tự tính :)

\(H=\frac{1}{1.2}-\frac{1}{1.2.3}+\frac{1}{2.3}-\frac{1}{2.3.4}+...+\frac{1}{99.100}-\frac{1}{99.100.101}\)

\(=\left(\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{99.100}\right)-\left(\frac{1}{1.2.3}+\frac{1}{2.34}+...+\frac{1}{99.100.101}\right)\)

\(=\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{99}-\frac{1}{100}\right)-\frac{1}{2}\left(\frac{2}{1.2.3}+\frac{2}{2.3.4}+...+\frac{2}{99.100.101}\right)\)

\(=\left(1-\frac{1}{100}\right)-\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{99.100}-\frac{1}{100.101}\right)\)

\(=\frac{99}{100}-\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{100.101}\right)=\frac{99}{100}-\frac{1}{2}.\frac{5049}{10100}=\frac{99}{100}-\frac{5049}{20200}=\frac{14949}{20200}\)

1/1x2x3+1/2x3x4+...1/118x19x20<1/4 <--- cái này đề sai ở 1/118x19x20 phải là 1/18x19x20

đem mớ này nhồi vào đầu rồi đầy quá đứt mạch máu não , tử vong tại chỗ

ta có:
4s=1.2.3.(4-0)+2.3.4.(5-1)+3.4.5.(6-2)+.........+k(k+1)(k+2)((k+3)-(k-1))
4s=1.2.3.4-1.2.3.0+2.3.4.5-1.2.3.4+3.4.5.6-2.3.4.5+........+k(k+1)(k+2)(k+3)-(k-1)k(k+1)(k+2)
4s=k(k+1)(k+2)(k+3)
ta biết rằng tích 4 số tự nhiên liên tiếp khi cộng thêm 1 luôn là 1 số chính phương
=>4s+1 là 1 số chính phương