Question

1/a(a+1)+1/(a+1)(a+2)+1/(a+2)(a+3 )+……+1/(a+2009)(a+2010)

Answer 1

令10^x=a
x=lga
所以f(a)=lga-2
f(x)=lgx-2

1000x-383.5/x=4.7
1000x^2-4.7x-383.5=0
求根公式
x=(47±√153402209)/20000
f(x+1)=lg(x+1)-2

f(2)=lg2-2

Answer 2

1/a(a+1)+1/(a+1)(a+2)+1/(a+2)(a+3 )+……+1/(a+2009)(a+2010)
=1/a-1/(a+1)+1/(a+1)-1/(a+2)+1/(a+2)-1/(a+3 )+……+1/(a+2009)-1/(a+2010)
=1/a-1/(a+2010)
=2010/[a(a+2010)]