Ta có :

\(S=\dfrac{2}{1.2.3}+\dfrac{2}{2.3.4}+\dfrac{2}{3.4.5}+...............+\dfrac{2}{2009.2010.2011}\)

\(S=\dfrac{1}{1.2}-\dfrac{1}{2.3}+\dfrac{1}{2.3}-\dfrac{1}{3.4}+.........+\dfrac{1}{2009.2010}-\dfrac{1}{2010.2011}\)

\(S=\dfrac{1}{1.2}-\dfrac{1}{2010.2011}\)

\(S=\dfrac{1}{2}-\dfrac{1}{4042110}\) \(< \dfrac{1}{2}\)

\(\Rightarrow S< Q\)

s=1/1*2-1/2*3+1/2*3-1/3*4+....+1/2009*2010-1/210*2011

 =1/1*2-1/2010*2011

<1/1*2

\(S=\frac{2}{1\cdot2\cdot3}+\frac{2}{2\cdot3\cdot4}+\frac{2}{3\cdot4\cdot5}+...+\frac{2}{2009\cdot2010\cdot2011}\)

\(S=\frac{1}{1\cdot2}-\frac{1}{2\cdot3}+\frac{1}{2\cdot3}-\frac{1}{3\cdot4}+\frac{1}{3\cdot4}-\frac{1}{4\cdot5}+...+\frac{1}{2009\cdot2010}-\frac{1}{2010\cdot2011}\)

\(S=\frac{1}{1\cdot2}-\frac{1}{2010\cdot2011}\)

\(S=\frac{1}{2}-\frac{1}{2010\cdot2011}< \frac{1}{2}\)

=> S < P

\(S=\dfrac{2}{1.2.3}+\dfrac{2}{2.3.4}+\dfrac{2}{3.4.5}+...+\dfrac{2}{2009.2010.2011}\)

\(=\dfrac{1}{1.2}-\dfrac{1}{2.3}+\dfrac{1}{2.3}-\dfrac{1}{3.4}+\dfrac{1}{3.4}-\dfrac{1}{4.5}+...+\dfrac{1}{2009.2010}-\dfrac{1}{2010.2011}\)

\(=\dfrac{1}{1.2}-\dfrac{1}{2010.2011}\)

\(=\dfrac{1}{2}-\dfrac{1}{4042110}< \dfrac{1}{2}\)

\(\Rightarrow\) \(S< P\)

Vậy \(S< P\)

Tổng quát: \(\frac{2}{\left(a-1\right)a\left(a+1\right)}=\frac{1}{\left(a-1\right).a}-\frac{1}{a\left(a+1\right)}\)

Ta có: \(S=\frac{2}{1.2.3}+\frac{2}{2.3.4}+.....+\frac{2}{2013.2014.2015}\)

\(S=\left(\frac{1}{1.2}-\frac{1}{2.3}\right)+\left(\frac{1}{2.3}-\frac{1}{3.4}\right)+.....+\left(\frac{1}{2013.2014}-\frac{1}{2014.2015}\right)\)

\(S=\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+....+\frac{1}{2013.2014}-\frac{1}{2014.2015}\)

\(S=\frac{1}{1.2}-\frac{1}{2014.2015}=\frac{1}{2}-\frac{1}{2014.2015}<\frac{1}{2}\)

Vậy....................

S=(2/1.2-2/2.3)+(2/2.3-2/3.4)+(2/3.4-2/4.5)+...........+(2/2013.2014-2/2014-2/2015)

S=(2/1.2-2/2014.2015):2

S=1-2/2014.2/2015

--> S>1/2

giải thích hộ chả hiểu

a) \(A=2^{2010}-2^{2009}-2^{2008}-...-2-1\)

\(A=2^{2010}\left(2^{2009}+2^{2008}+...+2+1\right)\)

Đặt \(\text{A = 1 + 2 + . . . + 2^{2008} + 2^{2009}}\)

\(\text{⇒ 2 A = 2 + 2 2 + . . + 2^{2010}}\)

⇒ \(A=2^{2010}-1\)

⇒ \(A=2^{2010}-\left(2^{2010}-1\right)\)

⇒ \(A=1\)

b) \(B=2072\)

c) \(\dfrac{4949}{19800}\)

Xin lỗi mình không có nhiều thời gian để giải thích trên đây á nên tạm gửi ảnh mình tạo nhé . Học tốt !

1/2-1/2010.2011

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