数列{An}:A(n+1)=3an-3an^2,a1=1/2,求证2/3

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数列{An}:A(n+1)=3an-3an^2,a1=1/2,求证2/3数列{An}:A(n+1)=3an-3an^2,a1=1/2,求证2/3数列{An}:A(n+1)=3an-3an^2,a1=1

数列{An}:A(n+1)=3an-3an^2,a1=1/2,求证2/3
数列{An}:A(n+1)=3an-3an^2,a1=1/2,求证2/3

数列{An}:A(n+1)=3an-3an^2,a1=1/2,求证2/3
【将数列看成离散型函数】
a[n+1]=3*a[n]-3*a[n]^2
a[n+2]=3*a[n+1]-3*a[n+1]^2=9a[n]-9a[n]^2-27(a[n]-a[n]^2)^2=9a[n](1-a[n])(3a[n]^2-3a[n]+1)
a[n+2]-a[n]=a[n](2-3a[n])^3
【一解】:
将a[n+2]看成a[n]的函数f(x),即f(x)=9x(1-x)(3x^2-3x+1)
f’(x)=-108x^3+162x^2-72x+9=9(1-2x)(6x^2-6x+1)
f’’(x)=-324x^2+324x-72=36(3x-2) (3x-1)
由f’(x)=0得f(x)的极值点:x1=(3-√3)/63/4
且f’’(x1)=f’’(x2)0,故f(x1)和f(x2)为极大值,f(x0)为极小值;
f(2/3)=2/3;f(3/4)=189/256;
因此,当2/31时,2/3