Question

已知数列｛an｝中,a1=5/6,且对任意正整数n都有a(n+1)=1/3an+(1/2)^(n+1).数列｛bn｝对任

Answer 1

(1)bn=a(n+1)-an/3=(1/2)^(n+1)

b(n+1)=a(n+2)-a(n+1)/3=(1/2)^(n+2)

b(n+1)/bn=1/2

等比数列

(2)a(n+1)-3*(1/2)^(n+1)=an/3-2*(1/2)^(n+1)=1/3*(an-3*(1/2)^n)

an-3*(1/2)^n为等比数列

an-3*(1/2)^n=(1/3)^(n-1)

从而an=-2*(1/3)^n+3*(1/2)^n (n>=2)

a1也符合故此即为通项