Nemeth, LaszloSzalay, Laszlo2024-11-042024-11-042018-01-011212-5059https://doi.org/10.5817/AM2018-3-177https://hdl.handle.net/11452/47407Let F-n denote the nth term of the Fibonacci sequence. In this paper, we investigate the Diophantine equation F-1(p) + 2F(2)(p) + . . . + kF(k)(p) = F-n(q) in the positive integers k and n, where p and q are given positive integers. A complete solution is given if the exponents are included in the set {1, 2}. Based on the specific cases we could solve, and a computer search with p, q, k <= 100 we conjecture that beside the trivial solutions only F-8 = F-1 + 2F(2 )+ 3F(3 )+ 4F(4), F-4(2 )= F-1 + 2F(2) + 3F(3), and F-4(3) = F-1(3)+ 2F(2)(3 )+ 3F(3)(3) satisfy the title equation.eninfo:eu-repo/semantics/closedAccessConsecutive fibonacci numbersPowersSequenceSumFibonacci sequenceDiophantine equationScience & technologyPhysical sciencesMathematicsArticle00044110040000517718854310.5817/AM2018-3-177