求答案:已知数列{an}满足a1=1,a2=3,若数列{an·an+1}是公比为2的等比数列,求通项an

Question

求答案:已知数列{an}满足a1=1,a2=3,若数列{an·an+1}是公比为2的等比数列,求通项an

Answer 1

{a(n)a(n+1)}为首项为 a(1)a(2) = 3,公比为2的等比数列.
a(n)a(n+1) = 3*2^(n-1).
a(2n-1)a(2n) = 3*2^(2n-1-1) = 3*4^(n-1).
a(2n)a(2n+1) = 3*2^(2n-1) = 6*4^(n-1).
2 = [6*4^(n-1)]/[3*4^(n-1)] = [a(2n)a(2n+1)]/[a(2n-1)a(2n)] = a(2n+1)/a(2n-1),
a(2n+1) = 2a(2n-1).
{a(2n-1)}是首项为a(1)=1,公比为2的等比数列.
a(2n-1)=2^(n-1).
a(2n) = [3*4^(n-1)]/a(2n-1) = [3*4^(n-1)]/2^(n-1) = 3*2^(n-1).
{a(n)}的通项公式为,
a(2n-1)= 2^(n-1),
a(2n) = 3*2^(n-1).