数列an中,a1=2,an=2a（n-1）+3的n次方（n属于N*,且大于等于2）,求an

题目

数列an中,a1=2,an=2a（n-1）+3的n次方（n属于N*,且大于等于2）,求an

答案

a(n+1)=2a(n) + 3^(n+1),
a(n+1)/3^(n+1) = (2/3)[a(n)/3^n] + 1,
a(n+1)/3^(n+1) - 3 = (2/3)[a(n)/3^n] - 2 = (2/3)[a(n)/3^n - 3]
{a(n)/3^n - 3}是首项为a(1)/3 - 3 = 2/3 - 3 = -7/3,公比为2/3的等比数列.
a(n)/3^n - 3 = (-7/3)(2/3)^(n-1),
a(n) - 3^(n+1) = -7*2^(n-1),
a(n) = 3^(n+1) - 7*2^(n-1)