Question

求各位数学高人帮忙解个题，要有详细的步骤谢谢 1\1×3 + 1\2×4+ 1\3×5 …… +1\18×20

1\1×3 + 1\2×4+ 1\3×5 …… +1\18×20

Answer 1

解：
因为1/[n(n+2)]=1/2 × [(n+2)-n]/[n(n+2)]=1/2 × [1/n-1/(n+2)]
所以1\1×3 + 1\2×4+ 1\3×5 …… +1\18×20
=1/2×(1/1-1/3)+1/2×(1/2-1/4)+1/2×(1/3-1/5)+……+1/2×(1/18-1/20)
=1/2×(1/1-1/3+1/2-1/4+1/3-1/5+……+1/18-1/20)
=1/2×(1+1/2-1/19-1/20)
=1/2×531/380
=531/760

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Answer 2

下次可不能这么写,容易看错，要加括号的：1/(18*20)
先看这个：
1/(a*(a+2) )=1/2 * (1/a - 1/(a+2) )
原式=1/2[ (1/1 - 1/3)+(1/2 -1/4)+(1/3 - 1/5)...+(1/17 - 1/19)+(1/18 - 1/20) ]
(看第一和第三个括号。-1/3和1/3消去，同理，1/4...1/18也可消去)
=1/2[ 1/1+1/2-1/19-1/20 ]
=1/2[ 531/380 ]
=531/760

Answer 3

因为1/[n*(n+2)]=(1/2)*{(1/n)-[1/(n+2)]},
1/1*3+1/2*4+1/3*5+1/4*6+……+1/16*18+1/17*19+1/18*20
=(1/2)*(1-1/3+1/2-1/4+1/3-1/5+1/4-1/6+……+1/16-1/18+1/17-1/19+1/18-1/20)
=(1/2)*(1+1/2-1/19-1/20)
=(1/2)*(531/380)
=531/760