Ghi lộn đề thiếu thì phải. Hình như thiếu phân số 1/2011

A = 2²⁰⁰⁹ - 2²⁰⁰⁸ - 2²⁰⁰⁷ - ... - 2² - 2 - 1

A = 2²⁰⁰⁹ - (2²⁰⁰⁸ + 2²⁰⁰⁷ + ... + 2² + 2 + 1)

Đặt B = 2²⁰⁰⁸ + 2²⁰⁰⁷ + ... + 2² + 2 + 1

2B = 2²⁰⁰⁹ + 2²⁰⁰⁸ + ... + 2³ + 2² + 2

2B - B = (2²⁰⁰⁹ + 2²⁰⁰⁸ + ... + 2³ + 2² + 2) - (2²⁰⁰⁸ + 2²⁰⁰⁷ + ... + 2² + 2 + 1)

B = 2²⁰⁰⁹ - 1

=> A = 2²⁰⁰⁹ - (2²⁰⁰⁹ - 1) = 2²⁰⁰⁹ - 2²⁰⁰⁹ + 1 = 0 + 1 = 1

Gọi \(S=\frac{2009}{1}+\frac{2008}{2}+...+\frac{1}{2009}\)

\(\Rightarrow S=\frac{2010-1}{1}+\frac{2010-2}{2}+...+\frac{2010-2009}{2009}\)

\(\Rightarrow S=2010-1+\frac{2010}{2}-1+...+\frac{2010}{2009}-1\)

\(\Rightarrow S=2010+\frac{2010}{2}+...+\frac{2010}{2009}-\left(1+1+..+1\right)\)

\(\Rightarrow S=2010+\frac{2010}{2}+...+\frac{2010}{2009}-2009\)

\(\Rightarrow S=\frac{2010}{2}+\frac{2010}{3}+...+\frac{2010}{2009}+1\)

\(\Rightarrow S=\frac{2010}{2}+\frac{2010}{3}+..+\frac{2010}{2009}+\frac{2010}{2010}\)

\(\Rightarrow S=2010\left(\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2010}\right)\)

Khi đó \(A=\frac{\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2010}}{2010\left(\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2010}\right)}=\frac{1}{2010}\)

A = 2²⁰⁰⁹ - 2²⁰⁰⁸ - 2²⁰⁰⁷ - .... - 2² - 2 - 1

= 2²⁰⁰⁹ - ( 2²⁰⁰⁸ + 2²⁰⁰⁷ + .... + 2² + 2 + 1 )

Đặt B = 1 + 2 + 2² + .... + 2²⁰⁰⁸

2B = 2(1 + 2 + 2² + .... + 2²⁰⁰⁸ )

= 2 + 2² + 2³ + .... + 2²⁰⁰⁹

2B - B = ( 2 + 2² + 2³ + .... + 2²⁰⁰⁹ ) - ( 1 + 2 + 2² + .... + 2²⁰⁰⁸ )

B = 2²⁰⁰⁹ - 1

=> A = 2²⁰⁰⁹ - ( 2²⁰⁰⁹ - 1 ) = 1

Vậy A = 1

có nhầm đề không vậy phải là 2010-

help me

but i don't know

I don't know, too!!!!

\(C=\frac{\frac{1}{2008}-\frac{1}{2009}-\frac{1}{2010}}{\frac{5}{2008}-\frac{5}{2009}-\frac{5}{2010}}+\frac{\frac{2}{2007}-\frac{2}{2008}-\frac{2}{2009}}{\frac{3}{2007}-\frac{3}{2008}-\frac{3}{2009}}\)

\(=\frac{\frac{1}{2008}-\frac{1}{2009}-\frac{1}{2010}}{5.\left(\frac{1}{2008}-\frac{1}{2009}-\frac{1}{2010}\right)}+\frac{2.\left(\frac{1}{2007}-\frac{1}{2008}-\frac{1}{2009}\right)}{3.\left(\frac{1}{2007}-\frac{1}{2008}-\frac{1}{2009}\right)}\)

\(=\frac{1}{5}+\frac{2}{3}\)

\(=\frac{13}{15}\)

có : Q = [ 2 + 2^2 ] + [ 2^3 +2^4] + ... + [2^9 +  2^10]

Q = 2 [1+2] +2^3[1 +2]+ ...+ 2^9 [1+2]

Q = 2 . 3+2^3 .3 +... + 2^9 .3

Q = 3. [ 2 + 2^3 +... + 2^9]

Vậy Q chia hết cho 3

1.

\(1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{99}}+\frac{1}{2^{100}}+\frac{1}{2^{100}}\)

\(=1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{99}}+\left(\frac{1}{2^{100}}+\frac{1}{2^{100}}\right)\)

\(=1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{99}}+\frac{1}{2^{99}}\)

cứ làm như vậy ta được :

\(=1+1=2\)

2. Ta có :

\(\frac{2008+2009}{2009+2010}=\frac{2008}{2009+2010}+\frac{2009}{2009+2010}\)

vì \(\frac{2008}{2009}>\frac{2008}{2009+2010}\); \(\frac{2009}{2010}>\frac{2009}{2009+2010}\)

\(\Rightarrow\frac{2008}{2009}+\frac{2009}{2010}>\frac{2008+2009}{2009+2010}\)