1/2*4+1/4*6+1/6*8+...+1/2012*2014

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1/2*4+1/4*6+1/6*8+...+1/2012*20141/2*4+1/4*6+1/6*8+...+1/2012*20141/2*4+1/4*6+1/6*8+...+1/2012*2014=

1/2*4+1/4*6+1/6*8+...+1/2012*2014
1/2*4+1/4*6+1/6*8+...+1/2012*2014

1/2*4+1/4*6+1/6*8+...+1/2012*2014
=1/4(1/1×2+1/2×3+1/3×4+……+1/1006×1007)
=1/4(1-1/2+1/2-1/3+1/3-1/4+……+1/1006-1/1007)
=1/4(1-1/1007)
=1/4×1006/1007
=503/2104