AT-OEMO-FED1 2005 Problem 3

For 3 real numbers $a,b,c$ let $s_n=a^{n}+b^{n}+c^{n}$ . It is known that $s_1=2$ , $s_2=6$ and $s_3=14$ . Prove that for all natural numbers $n>1$ , we have $|s^2_n-s_{n-1}s_{n+1}|=8$