数列an=2a(n-1)+2^n-1,a4=81,若{(an+p)/2^n}为等差数列,求实数p的值；

题目

数列an=2a(n-1)+2^n-1,a4=81,若{(an+p)/2^n}为等差数列,求实数p的值；

答案

an=2an-1 +2^(n-1)
an/2^n=an-1/2^(n-1)+1
设bn=an/2^n
bn=bn-1+1
bn-bn-1=1
bn等差数列
cn=a(n+p)/2^n p=0时,cn=bn,数列a(n+p)/2^n等差数列