On a Diophantine equation involving powers of Fibonacci numbers

This paper deals with the diophantine equation F 1 þ 2F p 2 þ þ kF p k 1⁄4 F n, an equation on the weighted power terms of Fibonacci sequence. For the exponents p; q 2 f1; 2g the problem has already been solved in ad hoc ways using the properties of the summatory identities appear on the left-hand side of the equation. Here we suggest a uniform treatment for arbitrary positive integers p and q which works, in practice, for small values. We obtained all the solutions for p; q 10 by testing the new approach.