已知:数列{an }满足a1+2a2+2^2·a3+``````+2^n-1·an=n/2(n属于正整数)1.求数列{an }的通项公式2.若bn=n/an,求数列{bn }的前n项的和Sn.
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已知:数列{an }满足a1+2a2+2^2·a3+``````+2^n-1·an=n/2(n属于正整数)1.求数列{an }的通项公式2.若bn=n/an,求数列{bn }的前n项的和Sn.
已知:数列{an }满足a1+2a2+2^2·a3+``````+2^n-1·an=n/2(n属于正整数)
1.求数列{an }的通项公式
2.若bn=n/an,求数列{bn }的前n项的和Sn.
已知:数列{an }满足a1+2a2+2^2·a3+``````+2^n-1·an=n/2(n属于正整数)1.求数列{an }的通项公式2.若bn=n/an,求数列{bn }的前n项的和Sn.
1、∵a1+2a2+...+2^(n-1)*an=n/2 ①
a1+2a2+...+2^n*a(n+1)=(n+1)/2 ②
②-①得2^n*a(n+1)=1/2,∴a(n+1)=1/(2^(n+1))
∴an=1/2^n
2、∴bn=n*2^n
Sn=1*2+2*2^2+……+n*2^n ③
2Sn= 1*2^2+……+(n-1)*2^n+n*2^(n+1) ④
③-④得-Sn=2+2^2+……+2^n-n*2^(n+1)=2^(n+1)-2-n*2^(n+1)
∴Sn=(n-1)*2^(n+1)+2
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