Đặt \(A=\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{98.99.100}\)

                                  \(A=\frac{1}{2}.\left(\frac{2}{1.2.3}+\frac{2}{2.3.4}+...+\frac{2}{98.99.100}\right)\)

                                \(A=\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+\frac{1}{3.4}-\frac{1}{4.5}+...+\frac{1}{98.99}-\frac{1}{99.100}\right)\)

                               \(A=\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{99.100}\right)\)

                            \(A=\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{9900}\right)\)

                           \(A=\frac{1}{2}.\left(\frac{4950-1}{9900}\right)=\frac{1}{2}.\frac{4949}{9900}=\frac{4949}{19800}\)

                         Ủng hộ mk nha!!

A = \(\frac{2}{1.2.3}\) + \(\frac{2}{2.3.4}\) + ... + \(\frac{2}{98.99.100}\)

A = \(\frac12\).(\(\frac{2}{1.3}\)) + \(\frac13\).(\(\frac{2}{2.4}\)) + ... + \(\frac{1}{99}\).(\(\frac{2}{98.100}\))

A = \(\frac12\).(\(\frac11-\frac13\)) + \(\frac13\).(\(\frac12-\frac14\)) + ...+ \(\frac{1}{99}\).(\(\frac{1}{98}-\frac{1}{100}\))

A = \(\frac{1}{1.2}\) - \(\frac{1}{2.3}\) + \(\frac{1}{2.3}\) - \(\frac{1}{3.4}\) + ...+\(\frac{1}{98.99}\) - \(\frac{1}{99.100}\)

A = \(\frac12-\frac{1}{9900}\)

A = \(\frac{4949}{9900}\)

= 1/2.(2/1.2.3+2/2.3.4+.....+2/50.51.52

=1/2.(1/1.2-1/2.3+1/2.3-1/3.4+....+1/50.51-1/51.52

=1/2.(1/1.2-1/51.52)

=1/2.(1/2-1/2652)

=1/2.1325/2652

=1325/5304

A=1/1.2-1/2.3+1/2.3-1/3.4+1/3.4-1/4.5+...+1/50.51-1/51.52

A=1/1.2-1/51.52

phần còn lại tự giải nhé

\(B=\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{48.49}-\frac{1}{49.50}\right)\)

\(=\frac{1}{2}\left(\frac{1}{2}-\frac{1}{49.50}\right)\)

Đến đây bạn tự tính nhé

\(B=\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{48.49.50}\)

\(B=\frac{1}{2}\cdot\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{48.49}-\frac{1}{49.50}\right)\)

\(B=\frac{1}{2}\cdot\left(\frac{1}{1.2}-\frac{1}{49.50}\right)\)

\(B=\frac{1}{2}\cdot\left(\frac{1}{2}-\frac{1}{2450}\right)\)

\(B=\frac{1}{2}\cdot\frac{612}{1225}=\frac{306}{1225}\)

Vậy \(B=\frac{306}{1225}\)

\(\frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+...+\frac{1}{98.99.100}\)

\(=\frac{1}{2}.\left(\frac{2}{1.2.3}+\frac{2}{2.3.4}+\frac{2}{3.4.5}+...+\frac{2}{98.99.100}\right)\)

\(=\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+\frac{1}{3.4}-\frac{1}{4.5}+...+\frac{1}{98.99}-\frac{1}{99.100}\right)\)

\(=\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{99.100}\right)\)

\(=\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{9900}\right)\)

\(=\frac{1}{2}.\frac{4949}{9900}\)

\(=\frac{4949}{19800}\)

Ai hỏi

\(2A=\frac{2}{1.2.3}+\frac{2}{2.3.4}+\frac{2}{3.4.5}+...+\frac{2}{98.99.100}\)

\(2A=\frac{3-1}{1.2.3}+\frac{4-2}{2.3.4}+\frac{5-3}{3.4.5}+...+\frac{100-98}{98.99.100}\)

\(2A=\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+\frac{1}{3.4}-\frac{1}{4.5}+...+\frac{1}{98.99}-\frac{1}{99.100}\)

\(2A=\frac{1}{2}-\frac{1}{99.100}=\frac{49}{99.100}\Rightarrow A=\frac{49}{2.99.100}\)

Câu 2:

2Q = \(\frac{2}{1.2.3}\) + \(\frac{2}{2.3.4}\) + ... + \(\frac{2}{98.99.100}\)

2Q = \(\frac12\).(\(\frac{2}{1.3}\)) + \(\frac13\).(\(\frac{2}{2.4}\)) + ... + \(\frac{1}{99}\).(\(\frac{2}{98.100}\))

2Q = \(\frac12\).(\(\frac11-\frac13\)) + \(\frac13\).(\(\frac12-\frac14\)) + ...+ \(\frac{1}{99}\).(\(\frac{1}{98}-\frac{1}{100}\))

2Q = \(\frac{1}{1.2}\) - \(\frac{1}{2.3}\) + \(\frac{1}{2.3}\) - \(\frac{1}{3.4}\) + ...+\(\frac{1}{98.99}\) - \(\frac{1}{99.100}\)

2Q = \(\frac12-\frac{1}{9900}\)

2Q = \(\frac{4949}{9900}\)

Q = \(\frac{4949}{9900}\) : 2

Q = \(\frac{4949}{19800}\)

Câu 1:

A = \(\frac14+\frac18+\frac{1}{16}+..+\frac{1}{128}\)

2A = \(\frac12+\frac14+\frac18+\cdots+\frac{1}{64}\)

2A - A = \(\frac12+\frac14+\frac18+\cdots+\frac{1}{64}\) - \(\frac14-\frac15-\frac{1}{16}-\ldots\frac{1}{128}\)

A = (\(\frac12-\frac{1}{128})+\left(\frac14-\frac14)+..+\left(\frac{1}{64}-\frac{1}{64}\right)\right.\)

A = \(\frac{64}{128}-\frac{1}{128}\) + 0 + 0+..+0

A = \(\frac{63}{128}\)

\(A=2.\left(\frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+...+\frac{1}{98.99.100}\right)\)

\(A=2.\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{98.99}-\frac{1}{99.100}\right)\)

\(A=2.\left(\frac{1}{1.2}-\frac{1}{99.100}\right)\)

\(A=2\cdot\frac{4949}{9900}=\frac{4949}{4950}\)

\(2S=\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{99.100}\)

\(2S=\frac{1}{2}-\frac{1}{9900}\)

\(2S=\frac{4949}{9900}\)

\(S=\frac{4949}{19800}\)

Ta xét : \(\frac{1}{1.2}-\frac{1}{2.3}=\frac{2}{1.2.3}\)

\(\frac{1}{2.3}-\frac{1}{3.4}=\frac{2}{2.3.4}\)

...

\(\frac{1}{98.99}-\frac{1}{99.100}=\frac{2}{98.99.100}\)

Ta có : 2S = \(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{98.99}-\frac{1}{99.100}\)

=> 2S = \(\frac{1}{1.2}-\frac{1}{99.100}\)

=> 2S = \(\frac{4949}{9900}\)

=> S = \(\frac{4949}{19800}\)