In: Advanced Math
Consider n numbers x1, x2, . . . , xn laid out on a circle and some value α. Consider the requirement that every number equals α times the sum of its two neighbors. For example, if α were zero, this would force all the numbers to be zero.
(a) Show that, no matter what α is, the system has a solution.
(b) Show that if α = 1/2 , then the system has a nontrivial solution.
(c) Show that if α = − 1/2 , then there is a nontrivial solution if and only if n is even.