若数列{an}满足（an+2）/(an+1)+(an+1)/an＝k（k为常数）,则称数列{an}为等比和数列,k称为公比和,已知数列{an}是以3为公比和的等比和数列,其中a1=1,a2=2,则a2013=

Question

若数列{an}满足 （an+2）/ (an+1) + (an+1)/ an ＝k（k为常数）,
则称数列{an}为等比和数列,k称为公比和,已知数列{an}是以3为公比和的等比和数列,其中a1=1,a2=2,则a2013=

Answer 1

k=3
a(n+2)/a(n+1) + a(n+1)/an = 3
a(n+2)/a(n+1) = - a(n+1)/an +3
a(n+2)/a(n+1) -3/2 = - [a(n+1)/an -3/2 ]
=>{a(n+1)/an -3/2}是等比数列, q=-1
a(n+1)/an -3/2 = (-1)^(n-1).(a2/a1 -3/2)
= -(1/2)(-1)^n
a(n+1)/an = 3/2 -(1/2)(-1)^n
an/a(n-1) = 3/2 -(1/2)(-1)^(n-1)
ie
an = a(n-1) ; if n is odd
= 2a(n-1) ; if n is even
if n is odd
an = a(n-1)
=2a(n-2)
=2a(n-2)
a2013= 2^(1006) .a1
= 2^(1006)