已知数列{an},{bn}与函数f(x),g(x),x∈R满足条件:

题目

已知数列{an},{bn}与函数f(x),g(x),x∈R满足条件:
b1=b,an=f(bn)=g(b(n+1))(n∈N*)
若f(x)=tx+1(t≠0,t≠2),g(x)=2x,f(b)≠g(b),且lim(an)(n→∞)存在,求t的取值范围,并求lim(an)(n→∞)(用t表示).

答案

由条件知：tbn+1=2b(n+1),且t≠2.可得b(n+1)+1/(t-2)=(t/2)[bn+1/(t-2)].由f(b)≠g(b),t≠2,t≠0,可知b+1/(t-2)≠0,t/2≠0,所以{bn+1/(t-2)}是首项为b+1/(t-2),公比为t/2的等比数列.bn+1/(t-2)=[b+1/(t-2)](t/2)^(n-...