Question

观察下列一组等式,然后解答后面的问题:(根号2+1)(根号2-1)=1;(根号3+根号2)(根号3-根号2)=1;(根号4+根号3

观察下列一组等式,然后解答后面的问题:(根号2+1)(根号2-1)=1;(根号3+根号2)(根号3-根号2)=1;(根号4+根号3)(根号4-根号3)=1;(根号5+根号4)(根号5-根号4)=1........观察上面的规律,计算下列式子的值.
(1/根号2+1,+1/根号3+根号2,+1/根号4+根号3，+...+1/根号2012+根号2011)(根号2012+1)

Answer 1

(1/根号2+1,+1/根号3+根号2,+1/根号4+根号3，+...+1/根号2012+根号2011)(根号2012+1)
=[(√2-1)/(√2+1)(√2-1)+(√3-√2)/(√3+√2)(√3-√2)+....(√2012-√2011)/(√2012+√2011)(√2012-√2011)](√2012+1)
=(√2-1+√3-√2+√4-√3+....+√2012-√2011)(√2012+1）
=(√2012-1)(√2012+1)
=(√2012)²-1
=2012-1

=2011

Answer 2