1/1+2 + 1/1+2+3 + 1/1+2+3+4 + …… 1/1+2+……+199+200
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![1/1+2 + 1/1+2+3 + 1/1+2+3+4 + …… 1/1+2+……+199+200](/uploads/image/z/1131713-17-3.jpg?t=1%2F1%2B2+%2B+1%2F1%2B2%2B3+%2B+1%2F1%2B2%2B3%2B4+%2B+%E2%80%A6%E2%80%A6+1%2F1%2B2%2B%E2%80%A6%E2%80%A6%2B199%2B200)
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1/1+2 + 1/1+2+3 + 1/1+2+3+4 + …… 1/1+2+……+199+200
1/1+2 + 1/1+2+3 + 1/1+2+3+4 + …… 1/1+2+……+199+200
1/1+2 + 1/1+2+3 + 1/1+2+3+4 + …… 1/1+2+……+199+200
1+2+3+.+n=n(n+1)/2
1/(1+2+3+.+n)=2/n(n+1)
1/(1+2) + 1/(1+2+3) + 1/1+2+3+4 + …… 1/1+2+……+199+200
=2/2*3+2/3*4+2/4*5+.+2/200*201
=2(1/2*3+1/3*4+1/4*5+.+1/200*201)
=2(1/2-1/3+1/3-1/4+1/4-1/5+.+1/200-1/201)
=2*(1/2-1/201)
=2*1/2-2*1/201
=1-2/201
=199/201
10100
通式为2/n(n+1)=2(1/n-1/n+1)
所以原式=2(1/2-1/3+1/3-1/4.....+1/199-1/200+1/200-1/201)=1-2/201=199/201