Question

$sqrt[]{1+2005 sqrt[]{1+2004 sqrt[]{1+2003*2001}}}$

Answer 1

$sqrt{1 + 2005 sqrt{1+2004 sqrt{1 + 2003*2001}}} = \ = sqrt{1 + 2005 sqrt{1+2004 sqrt{1 + (2002+1)*(2002-1)}}}=\= sqrt{1 + 2005 sqrt{1+2004 sqrt{1 + 2002^2 - 1 }}} =\ = sqrt{1 + 2005 sqrt{1+2004 * 2002}} =\= sqrt{1 + 2005 * 2003} = 2004$

Answer 2

$sqrt{1+2005sqrt{1+2004sqrt{1+2003cdot2001}}}=\ =sqrt{1+2005sqrt{1+2004sqrt{1+2cdot2001+2001cdot2001}}}=\ =sqrt{1+2005sqrt{1+2004sqrt{1+2cdot1cdot2001+2001^2}}}=\ =sqrt{1+2005sqrt{1+2004sqrt{(1+2001)^2}}}=\ =sqrt{1+2005sqrt{1+2004sqrt{2002^2}}}=sqrt{1+2005sqrt{1+2004cdot2002}}=\ =sqrt{1+2005sqrt{1+2cdot2002+2002cdot2002}}=\ =sqrt{1+2005sqrt{1+2cdot1cdot2002+2002^2}}=sqrt{1+2005sqrt{(1+2002)^2}}=\ =sqrt{1+2005sqrt{2003^2}}=sqrt{1+2005cdot2003}=$

$=sqrt{1+2cdot2003+2003cdot2003}=sqrt{1+2cdot1cdot2003+2003^2}=\ =sqrt{(1+2003)^2}=sqrt{2004^2}=2004$