On 04/07/2016 06:55, Zak Seidov via math-fun wrote:
If a(n)=3*a(n-1)-a(n-2), then (a (n+1) - a (n))^2 = a (n)*(n + 1) - 1. How to prove/disprove it?
Well, it certainly can't be true in general, e.g. because shifting the whole sequence one place (i.e., renumbering) preserves the recurrence relation and the LHS of your equation but changes the RHS. If it's true for a particular sequence satisfying the recurrence, you can brute-force it by getting an explicit formula for the elements. (I wonder -- not having checked any actual numbers -- whether there's an "a" missing in your equation: a(n)*a(n+1), not a(n)*(n+1). In that case, it still can't be true in general because if (a(n)) is a solution to the recurrence then so is (2a(n)), and all terms of your equation other than the constant 1 are homogeneous of degree 2. And, again, if it's true for a particular sequence it won't be too hard to prove via an explicit formula.) -- g