1/2*4+1/4*6+1/6*8+.+1/2008*2010
= 1/4 * ( 1/1*2 + 1/2*3 + 1/3*4 + .+ 1/1004*1005 )
= 1/4 * ( 1-1/2 + 1/2-1/3 + 1/3-1/4 + .+ 1/1004-1/1005 )
= 1/4 * (1 - 1/1005)
= 1/4 * 1004/1005
= 251/1005