a) \(A=2^{100}-2^{99}-2^{98}-...-2^2-2^1\)( Có 2 câu nên mình tính nhanh luôn nhé )

\(\Leftrightarrow A=2^{100}-\left(2^1+2^2+2^3+...+2^{98}+2^{99}\right)\)

\(A=2^{100}-\left(2^{100}-2^1\right)=2^{100}-2^{100}+2=2\)

b) \(\frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+...+\frac{1}{36.37.38}+\frac{1}{37.38.39}\)

\(=\frac{3-1}{1.2.3}+\frac{4-2}{2.3.4}+\frac{5-3}{3.4.5}+...+\frac{38-36}{36.37.38}+\frac{39-37}{37.38.39}\)

\(=\left(\frac{3}{1.2.3}-\frac{1}{1.2.3}\right)+\left(\frac{4}{2.3.4}-\frac{2}{2.3.4}\right)+...+\left(\frac{39}{37.38.39}-\frac{37}{37.38.39}\right)\)

\(=\left(\frac{1}{2}-\frac{2}{3}\right)+\left(\frac{2}{3}-\frac{3}{4}\right)+\left(\frac{3}{4}-\frac{4}{5}\right)+...+\left(\frac{1}{37.38}-\frac{1}{38.39}\right)\)

\(=\frac{1}{2}-\frac{2}{3}+\frac{2}{3}-\frac{3}{4}+\frac{3}{4}-\frac{4}{5}+...+\frac{1}{37.38}-\frac{1}{38.39}\)

\(=\frac{1}{2}-\frac{1}{38.39}=\frac{741}{1482}-\frac{1}{1482}=\frac{740}{1482}=\frac{370}{741}\)

\(\frac{1}{1\cdot2\cdot3}+\frac{1}{2\cdot3\cdot4}+\frac{1}{3\cdot4\cdot5}+...+\frac{1}{37\cdot38\cdot39}\)

\(=\frac{1}{2}\left(\frac{1}{1\cdot2}-\frac{1}{2\cdot3}+\frac{1}{2\cdot3}-\frac{1}{3\cdot4}+\frac{1}{3\cdot4}-\frac{1}{4\cdot5}+...+\frac{1}{37\cdot38}-\frac{1}{38\cdot39}\right)\)

\(=\frac{1}{2}\left(\frac{1}{1\cdot2}-\frac{1}{38\cdot39}\right)\)

\(=\frac{1}{2}\left(\frac{1}{2}-\frac{1}{1482}\right)\)

\(=\frac{1}{2}\cdot\frac{370}{741}\)

\(=\frac{185}{741}\)

\(S=\dfrac{1}{1.2.3}+\dfrac{1}{2.3.4}+...+\dfrac{1}{37.38.39}\)

\(\Rightarrow S=\dfrac{1}{2}\left(\dfrac{2}{1.2.3}+\dfrac{2}{2.3.4}+...+\dfrac{2}{37.38.39}\right)\)

\(\Rightarrow S=\dfrac{1}{2}\left(\dfrac{1}{1.2}-\dfrac{1}{2.3}+\dfrac{1}{2.3}-\dfrac{1}{3.4}+...+\dfrac{1}{37.38}-\dfrac{1}{38.39}\right)\)

\(\Rightarrow S=\dfrac{1}{2}\left(\dfrac{1}{1.2}-\dfrac{1}{38.39}\right)\)

\(\Rightarrow S=\dfrac{1}{2}\left(\dfrac{1}{2}-\dfrac{1}{38.39}\right)\)

\(\Rightarrow S=\dfrac{1}{4}-\dfrac{1}{38.2.39}\)

Vậy...

\(S=\dfrac{1}{2}.\left(\dfrac{1}{1.2}-\dfrac{1}{2.3}+\dfrac{1}{2.3}-\dfrac{1}{3.4}+..+\dfrac{1}{37.38}-\dfrac{1}{38.39}\right)\)

\(S=\dfrac{1}{2}.\left(\dfrac{1}{1.2}-\dfrac{1}{38.39}\right)\)

\(S=\dfrac{185}{741}\)

mk nghĩ vậy bạn ạ, nếu sai đừng trách mk nha bạn

\(\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{37.38.39}\)

\(=\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{37.38}-\frac{1}{38.39}\right)\)

\(=\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{38.39}\right)\)

\(=\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{1482}\right)\)

\(=\frac{1}{2}.\left(\frac{741}{1482}-\frac{1}{1482}\right)\)

\(=\frac{1}{2}.\frac{740}{1482}\)

\(=\frac{185}{741}\)

Chúc bạn học tốt !!! 

* Công thức :  

\(\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{2.3}\right)=\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{6}\right)=\frac{1}{2}.\left(\frac{3}{6}-\frac{1}{6}\right)=\frac{1}{2}.\frac{2}{6}=\frac{1}{6}=\frac{1}{1.2.3}\)

VD ( dễ hiểu ) 

:)))) :>>> :+ := :- :

185/741

\(A=\dfrac{1}{1.2.3}+\dfrac{1}{2.3.4}+....+\dfrac{1}{37.38.39}\)

\(A=\dfrac{1}{2}.\left(\dfrac{2}{1.2.3}+\dfrac{2}{2.3.4}+....+\dfrac{2}{37.38.39}\right)\)\(A=\dfrac{1}{2}\left(\dfrac{3-1}{1.2.3}+\dfrac{4-2}{2.3.4}+....+\dfrac{39-37}{37.38.39}\right)\)

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