(1*1+2*2)/(1*2)+(2*2+3*3)/(2*3)+……+(2000*2000+2001*2001)/(2000*2001)
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![(1*1+2*2)/(1*2)+(2*2+3*3)/(2*3)+……+(2000*2000+2001*2001)/(2000*2001)](/uploads/image/z/3731115-3-5.jpg?t=%281%2A1%2B2%2A2%29%2F%281%2A2%29%2B%282%2A2%2B3%2A3%29%2F%282%2A3%29%2B%E2%80%A6%E2%80%A6%2B%EF%BC%882000%2A2000%2B2001%2A2001%29%2F%282000%2A2001%29)
(1*1+2*2)/(1*2)+(2*2+3*3)/(2*3)+……+(2000*2000+2001*2001)/(2000*2001)
(1*1+2*2)/(1*2)+(2*2+3*3)/(2*3)+……+(2000*2000+2001*2001)/(2000*2001)
(1*1+2*2)/(1*2)+(2*2+3*3)/(2*3)+……+(2000*2000+2001*2001)/(2000*2001)
(1*1+2*2)/(1*2)+(2*2+3*3)/(2*3)+……+(2000*2000+2001*2001)/(2000*2001)
=(1/2+2/1)+(2/3+3/2)+.(2000/2001+2001/2000)
=(1-1/2+1+1/1)+(1-1/3+1+1/2)+...(1-1/2001+1+1/2000)
=(2-1/2+1/1)+(2-1/3+1/2)+...(2-1/2001+1/2000)
=2*2000+1-1/2001
=4001-1/2001
=4000又2000/2001
可以观察其中任一项n,
(n*n+(n+1)*(n+1))/n(n+1)=n/(n+1)+(n+1)/n=1-1/(n+1)+1+1/n=2+1/n-1/(n+1)
所以(1*1+2*2)/(1*2)+(2*2+3*3)/(2*3)+……+(2000*2000+2001*2001)/(2000*2001)
=2+1-1/2+2+1/2-1/3+......+2+1/2000-1/2001
=2*2000+1-1/2001
=4001-1/2001
=800600/2001
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