The property is as defined as $a_{2n}= a_n b_n$. Where $a_n$ is the fibonacci sequence and $b_n$ is the sequence of lucas numbers. Fibonacci series is $a_1 = a_2 =1,a_3 = 2,a_n = a_{n-1}+a_{n-2}$ and lucas numbers are $b_1 = 1, b_n = b_{n-1}+b_{n-2}$ and $b_n= a_{n+1} + a_{n-1}$ for $n$ more than or equal to $2$.
You have to prove it using induction or strong induction.I have reached 'till say $a_{2n} a_n b_n$ then for $n+1$ \begin{align*} a_{2(n+1) }& =a_{2n +2} \\ & =b_{2n+1} - a_{2n} \\ & = b_{2n+1} - a_n b_n \end{align*} Now I am stuck.
