Tính nhanh giá trị các biểu thức: A= 1/2 + 1/2^2 +1/2^3 +...+1/2^2009 +1/2^2010

Ta có:

$\frac{2009-\frac{2009}{2001}-\frac{2009}{2002}-\frac{2009}{2003}-\frac{2009}{2004}}{2010-\frac{2010}{2001}-\frac{2010}{2002}-\frac{2010}{2003}-\frac{2010}{2004}}=\frac{2009(1-\frac{1}{2001}-\frac{1}{2002}-\frac{1}{2003}-\frac{1}{2004})}{2010(1-\frac{1}{2001}-\frac{1}{2002}-\frac{1}{2003}-\frac{1}{2004})}=\frac{2009}{2010}$

Tương tự:

$\frac{2009-\frac{2009}{2005}-\frac{2009}{2006}-\frac{2009}{2007}-\frac{2009}{2008}}{2010-\frac{2010}{2005}-\frac{2010}{2006}-\frac{2010}{2007}-\frac{2010}{2008}}=\frac{2009(1-\frac{1}{2005}-\frac{1}{2006}-\frac{1}{2007}-\frac{1}{2008})}{2010(1-\frac{1}{2005}-\frac{1}{2006}-\frac{1}{2007}-\frac{1}{2008})}=\frac{2009}{2010}$

=> $B=\frac{2009}{2010}:\frac{2009}{2010}=1$

đề phải là 2²⁰⁰⁹  và 2²⁰¹⁰ mới đúng

Ta có

$2A=1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{2008}}+\frac{1}{2^{2009}}$

=> $2A-A=1-\frac{1}{2^{2010}}$

=> $A=\frac{2^{2010}-1}{2^{2010}}$