用规律计算：1／1+√2+1／√2+√3+1／√3√4+、、、+1／√2008+√2009+1／√2009√2010+1／√2010+√2011根据(√2-1)*(√2+1)=1(√3-√2)*(√3+√2)=1(√2005-√2004)*(√2005+√2004)=1

Question

用规律计算：1／1+√2+1／√2+√3+1／√3√4+、、、+1／√2008+√2009+1／√2009√2010+1／√2010+√2011
根据(√2-1)*(√2+1)=1
(√3-√2)*(√3+√2)=1
(√2005-√2004)*(√2005+√2004)=1

Answer 1

由规律可知：
1／(1+√2) +1／(√2+√3)+1／(√3+√4)+...+1／(√2009+√2010)+1／(√2010+√2011)
=(√2 -1)／[(1+√2)(√2 -1)] +(√3-√2)／[(√2+√3)(√3-√2)]+(√4-√3)／[(√3+√4)(√4-√3)]+...
+(√2010-√2009)／[(√2009+√2010)(√2010+√2009)]
+(√2011-√2010)／[(√2010+√2011)(√2011+√2010)]
=(√2 -1)+(√3-√2)+(√4-√3)+...+(√2010-√2009)+(√2011-√2010)
=(√2011) -1