  Let $F_n$ denote the $n^{th}$ term of the Fibonacci sequence. In this paper,
we investigate the Diophantine equation $F_1^p+2F_2^p+\cdots+kF_{k}^p=F_{n}^q$
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 $p,q,k\le100$ 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.
