求一道数列求和求和Sn=1*2+2*2^n+3*2^n+...+n*2^n(求详细解答)

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求一道数列求和求和Sn=1*2+2*2^n+3*2^n+...+n*2^n(求详细解答)
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求一道数列求和求和Sn=1*2+2*2^n+3*2^n+...+n*2^n(求详细解答)
求一道数列求和
求和Sn=1*2+2*2^n+3*2^n+...+n*2^n(求详细解答)

求一道数列求和求和Sn=1*2+2*2^n+3*2^n+...+n*2^n(求详细解答)
原题应该是Sn=1*2^n+2*2^n+3*2^n+...+n*2^n吧,如果是这样的话解题步骤如下
Sn=(1+2+3+···+n)2^n
=n(n+1)/2x2^n
=n(n+1)2^(n-1)
如果是你给的题的话,解题步骤如下:
Sn=1*2+2*2^n+3*2^n+...+n*2^n
=2+(2+3+···+n)2^n
=2+(n+2)(n-1)/2x2^n
=2+(n+2)(n-1)2^(n-1)