若F(X)=(3X+2)/(X+2) 满足A1=0.5 A(n+1)=F(An) B(n)=1/(An +1) 求Bn的
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若F(X)=(3X+2)/(X+2) 满足A1=0.5 A(n+1)=F(An) B(n)=1/(An +1) 求Bn的通项公式
a(n+1)=(3*an+2)/(an+2)上式两边同加1的a(n+1)+1=(3*an+2)/(an+2)+1a(n+1)+1=4*(an+1)/an+2两边取倒数1/(a(n+1)+1)=(an+2)/4*(an+1) =(an+1+1)/4*(an+1) =1/4+(1/4)*(1/an+1)∵bn=1/(an+1)∴b(n+1)=1/4+(1/4)*bn上式可化为b(n+1)-1/3=(1/4)*(bn-1/3)所以数列{bn-1/3}是公比为1/4的等比数列bn-1/3=(b1-1/3)*(1/4)^(n-1) =(1/(0.5+1)-1/3)*(1/4)^(n-1) =(1/3)*(1/4)^(n-1)bn=(1/3)*(1/4)^(n-1)+1/3
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