令原式=Sn
an=(2n+1)/2^n (n为自然数，n=0,1,2...101)
a(n+1)=(2n+3)/2^(n+1)
a(n+1)-an=(2n+3-4n-2)/2^(n+1)=-(2n-1)/[2^(n-1)*4]=-a(n-1)/4
a(n+1)=an-a(n-1)/4=a(n-1)-a(n-1)/4-a(n-2)/4
=...
=a1-[a(n-1)+a(n-2)+...+a0]/4
=a1-Sn/4+an/4
Sn=4a1+an-4a(n+1)
=4*3/2+(2n+1)/2^n-2*(2n+3)/2^n
=6+(2n+1-4n-6)/2^n
=6-(2n+5)/2^n
将n=101代入
Sn=6-207/2^101
这是请教群里的一位家长做的,我也最不出.
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