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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 ) 

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

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

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

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

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

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

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

ta co tu suy nghi 

chuẩn CMNR ;+

Cách 1:

 A = \(\frac{2}{1.2.3}\)+ \(\frac{2}{2.3.4}\)+ \(\frac{2}{3.4.5}\)+ .... + \(\frac{2}{36.37.38}\)+ \(\frac{2}{37.38.39}\)

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

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

A = ( \(\frac{1}{1.2}\)- \(\frac{1}{2.3}\)) + (\(\frac{1}{2.3}\) -\(\frac{1}{3.4}\) ) + ... + ( \(\frac{1}{36.37}\)- \(\frac{1}{37.38}\)) + ( \(\frac{1}{37.38}\)- \(\frac{1}{38.39}\))

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

A = \(\frac{1}{1.2}\)+ 0 + 0 + 0 +... + 0 - \(\frac{1}{38.39}\)

A = \(\frac{1}{1.2}\)- \(\frac{1}{38.39}\)= \(\frac{741}{1482}\)- \(\frac{1}{1482}\)= \(\frac{740}{1482}\)=\(\frac{370}{741}\)

Ta có: S = 1.2 + 2.3 + 3.4 +.....+ 37.38 + 38.39

=> 3S = 1.2.(3 - 0) + 2.3.(4 - 1) + .... + 37.38.(39 - 36) + 38.39.(40 - 37)

=> 3S = 1.2.3 - 1.2.3 + 2.3.4 - 2.3.4 + .... + 38.39.40

=> 3S = 38.39.40

=> S = 38.39.40 / 3

=> S = 19760

3s=1*2*3+2*3*3+....+37*38*3+38*39*3

3S=1*2*(3-0) + 2*3*(4-1) + ...+37*38*(39-36) + 38*39*(40-37)

3S=1*2*3-1*2*0 + 2*3*4-1*2*3 + ...+38*39*40-37*38*39

3s=37*37*39-1*2*0

3S=37*38*39=54834

S=18278