Question

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

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

an.a(n+1) = (a1.a2)2^(n-1)
= 3.2^(n-1)
log<2>an + log<2>a(n+1) = n-1 + log<2>3
log<2>a(n+1) -(1/2)(n+1) + (1/2)(3/2 - log<2>3) = - [ log<2>an-(1/2)n +(1/2)(3/2 - log<2>3) ]
=>{ log<2>an-(1/2)n +(1/2)(3/2 - log<2>3) }是等比数列, q=-1
log<2>an-(1/2)n +(1/2)(3/2 - log<2>3) = (-1)^(n-2) . [ log<2>a2-(1/2)(2) +(1/2)(3/2 - log<2>3) ]
= (-1)^n [ (1/2)log<2>3 - (1/4)]
log<2>an = (1/2)n -(1/2)(3/2 - log<2>3) + (-1)^n [ (1/2)log<2>3 - (1/4)]
an = 2^{(1/2)n -(1/2)(3/2 - log<2>3) + (-1)^n [ (1/2)log<2>3 - (1/4)]}

Answer 2