已知|ab-2|与|a-1|互为相反数求值1/ab+1/(a+1)(b+1)+1/(a+2)(b+2)+.1/(a+2006)(b+2006)已知|ab-2|与|a-1|互为相反数求值1/ab+1/(a+1)(b+1)+1/(a+2)(b+2)+.1/(a+2006)(b+2006)

Question

已知|ab-2|与|a-1|互为相反数 求值 1/ab+1/(a+1)(b+1)+1/(a+2)(b+2)+.1/(a+2006)(b+2006)
已知|ab-2|与|a-1|互为相反数求值 1/ab+1/(a+1)(b+1)+1/(a+2)(b+2)+.1/(a+2006)(b+2006)

Answer 1

绝对值项恒非负,只有0的相反数仍为非负数.
ab-2=0
a-1=0
解得a=1 b=2
b=a+1
1/(ab)+1/[(a+1)(b+1)]+1/[(a+2)(b+2)]+...+1/[(a+2006)(b+2006)]
=1/[a(a+1)]+1/[(a+1)(a+2)]+1/[(a+2)(a+3)]+...+1/[(a+2006)(a+2007)]
=1/a -1/(a+1)+1/(a+1)-1/(a+2)+1/(a+2)-1/(a+3)+...+1/(a+2006)-1/(a+2007)
=1/a -1/(a+2007)
=1/1 -1/2008
=2007/2008