\(\frac{2011x2010-1}{2009x2011+2010}=\frac{2011x\left(2009+1\right)-1}{2009x2011+2010}\)

\(=\frac{2011x2009+2011-1}{2009x2011+2010}=\frac{2011x2009+2010}{2009x2011+2010}=1\)

b.  Gọi tổng trên là A

=> A=10,11+11,12+12,13+....+97,98+98,99+99,100

=> A-99,1=10,11+11,12+12,13+.....+97,98+98,99

SSH của A-99,1 là

(98,99-10,11):1,01+1=89(SH)

Giá trị của A-99,1 là

\(\frac{89}{2}.\left(10,11+98,99\right)=4854,95\)

Vì A-99.1=4854,95

=> A=4854,95+99,1

=>A=4954.05

****

a) \(\frac{2011.2010-1}{2009.2011+2010}=\frac{2011.2009+2011-1}{2009.2011+2010}=\frac{2011.2009+2010}{2009.2011+2010}=1\)

. là nhân nha

a) Tài=2^0+2^1+2^2+2^3+.......+2^2010 và bạn =2^2011-1

T = 2^0 + 2^1 + ... + 2^2010

2T = 2 + 2^2 + ... + 2^2011

2T - T = 2 + 2^2 + ... + 2^2011 - 2^0 - 2^1 - ... - 2^2010

T = (2 - 2^1) + (2^2 - 2^2)+..+(2^2010-2^2010)+(2^2011-2^0)

T = 0 + 0 + .. +0+ 2^2011 - 1

T = 2^2011 - 1 = bạn

Vậy Tài = Bạn

b) Tài= 2009x2011 và bạn=2010^2

T = 2009 x 2011

T = 2009 x (2010 + 1)

T = 2009 x 2010 + 2009

B = 2010 x (2009 + 1)

B = 2009 x 2010 + 2010 > 2009 x 2010 + 2009 = T

Tài < Bạn

$\frac{\frac{2010}{2011}}{\frac{2012}{2013}}+\frac{\frac{2011}{2012}}{\frac{2013}{2014}}+\frac{\frac{2012}{2013}}{\frac{2014}{2015}}$

$\frac{\frac{2010}{2011}}{\frac{2012}{2013}}+\frac{\frac{2011}{2012}}{\frac{2013}{2014}}+\frac{\frac{2012}{2013}}{\frac{2014}{2015}}$

$\frac{\frac{2010+2011+2012}{2011+2012+2013}}{\frac{2012+2013+2014}{2013+2014+2015}}$

$\frac{\frac{2010}{2011}+\frac{2011}{2012}+\frac{2012}{2013}}{\frac{2012+2013+2014}{2013+2014+2015}}$

$\frac{\frac{2010+2011+2012}{2011+2012+2013}}{\frac{2012}{2013}+\frac{2013}{2014}+\frac{2014}{2015}}$

dễ ợt nhưng éo biết làm thông cảm nha

10.

Sửa lại đề :Cho \(P=\dfrac{2009}{2010}+\dfrac{2010}{2011}+\dfrac{2012}{2013}+\dfrac{2013}{2009}\).Chứng tỏ rằng P<5.

\(P=\dfrac{2009}{2010}+\dfrac{2010}{2011}+\dfrac{2012}{2013}+\dfrac{2013}{2009}\)

\(P=\dfrac{2011}{2012}\)

\(\Rightarrow P< 5\)

\(P=\frac{2010}{2011}+\frac{2011}{2012}+\frac{2012}{2013}=\frac{2010}{8144863716}+\frac{2011}{8144863716}+\frac{2012}{8144863716}\)

\(=\frac{6033}{8144863716}=\frac{1}{1350052}\)

\(Q=2010+2011+\frac{2012}{2011}+2012+2013\)

\(=2010+2011+2012+2013+\frac{2012}{2011}\)

\(=8046+\frac{2012}{2011}=\frac{8046}{1}+\frac{2012}{2011}\)

\(=\frac{16180506}{2011}+\frac{2012}{2011}=\frac{16182518}{2011}\)

ai giải dùm vs

\(\frac{2010+2011+2012}{2011+2012+2013}=\frac{2010}{2011+2012+2013}+\frac{2011}{2011+2012+2013}+\frac{2012}{2011+2012+2013}\)

Vì \(\frac{2010}{2011+2012+2013}<\frac{2010}{2011};\frac{2011}{2011+2012+2013}<\frac{2011}{2012};\frac{2012}{2011+2012+2013}<\frac{2012}{2013}\)

nên phép dưới nhỏ hơn phép trên

Ta có: \(S=\frac{2010}{2011}+\frac{2011}{2012}+\frac{2012}{2013}+\frac{2013}{2014}\)    

\(< \frac{2011}{2011}+\frac{2012}{2012}+\frac{2013}{2013}+\frac{2014}{2014}\)\(=1+1+1+1=4\)

Vậy S < 4

do linh hình như sai đề

\(\frac{2009}{2010}<\frac{2010}{2011}<\frac{2011}{2012}<\frac{2012}{2013}<\frac{2013}{2014}\) 

xin lỗi bạn mik ko có thời gian ghi cách làm