Question

对于任意非零实数a,b,定义运算"☆"如下:a☆b=a-b/2ab,求2☆1+3☆2+4☆3+···+2010☆2009的值.

Answer 1

a☆b=(a-b)/2ab=a/2ab-b/2ab=1/2b-1/2a=1/2(1/b-1/a)
2☆1+3☆2+4☆3+···+2010☆2009
=1/2(1/1-1/2)+1/(1/2-1/3)+1/2(1/3-1/4)+……+1/2(1/2009-1/2010)
=1/2(1-1/2010)
=2009/4020

Answer 2

2☆1+3☆2+4☆3+···+2010☆2009
=(2-1)/(2×2×1)+(3-2)/(2×3×2)+(4-3)/(2×4×3)+······+(2010-2009)/(2×2010×2009)
=1/(2×2×1)+1/(2×3×2)+1/(2×4×3)+······+1/(2×2010×2009)
=1/2[1/(2×1)+1/(3×2)+1/(4×3)+······+1/(2010×2009)]
=1/2(1/1-1/2+1/2-1/3+1/3-1/4+······+1/2009-1/2010)
=1/2(1-1/2010)
=1/2×2009/2010
=2009/4020