求和:1/1*3+1/2*4+1/3*5+.+1/n(n+1) 1/1*4+1/4*7+1/7*10+.+1/(n+2)
来源:学生作业帮 编辑:大师作文网作业帮 分类:数学作业 时间:2024/09/22 04:10:52
求和:1/1*3+1/2*4+1/3*5+.+1/n(n+1) 1/1*4+1/4*7+1/7*10+.+1/(n+2)(n+1)
1:Sn=1/1*3+1/2*4+1/3*5+.+1/n*(n+2)
= 1/2*(1-1/3) + 1/2*(1/2 - 1/4) + ...+ 1/2*(1/n - 1/(n+2))
= 1/2 ( 1+ 1/2 - 1/(n+1) - 1/(n+2))
2:1/1*4+1/4*7+1/7*10+.+1/(3n-2)*(3n+1)
=(1/3)*[3/1*4+3/4*7+3/7*10+.+3/(3n-2)*(3n+1)]
=(1/3)*[(1/1-1/4)+(1/4-1/7)+(1/7-1/10)+...+(1/(3n-2)-1/(3n+1))]
=(1/3)*(1-1/(3n+1))
=n/(3n+1)
= 1/2*(1-1/3) + 1/2*(1/2 - 1/4) + ...+ 1/2*(1/n - 1/(n+2))
= 1/2 ( 1+ 1/2 - 1/(n+1) - 1/(n+2))
2:1/1*4+1/4*7+1/7*10+.+1/(3n-2)*(3n+1)
=(1/3)*[3/1*4+3/4*7+3/7*10+.+3/(3n-2)*(3n+1)]
=(1/3)*[(1/1-1/4)+(1/4-1/7)+(1/7-1/10)+...+(1/(3n-2)-1/(3n+1))]
=(1/3)*(1-1/(3n+1))
=n/(3n+1)
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