It is known that f (x) is a function defined on R which is not always zero. For any x, y ∈ R, f (x · y) = XF (y) + YF (x) holds. & nbsp; sequence {an} satisfies an = f (2n) (n ∈ n *), and A1 = 2. Then the general term formula of sequence an=______ ．

It is known that f (x) is a function defined on R which is not always zero. For any x, y ∈ R, f (x · y) = XF (y) + YF (x) holds. & nbsp; sequence {an} satisfies an = f (2n) (n ∈ n *), and A1 = 2. Then the general term formula of sequence an=______ ．

Since an = f (2n), then an + 1 = f (2n + 1) and A1 = 2 = f (2) ∵ for any x, y ∈ R, there is f (x · y) = XF (y) + YF (x) ∵ such that x = 2n, y = 2, then f (2n + 1) = 2NF (2) + 2F (2n) ∵ an + 1 = 2An + 2 × 2n ∵ an + 12n + 1 − an2n = 1 ∵ sequence {an2n} is an arithmetic sequence with A12 = 1 as the first term tolerance 1 ∵ an2n = 1 + & nbsp; (n − 1) × 1 = n ∵ an = n2n