On $k$-Fibonacci sums by matrix methods

In this paper, some $k$-Fibonacci and $k$-Lucas with arithmetic indexes sums are derived by using the matrices $R_{a}=\left[ \begin{array}{lr} L_{k,a} & -(-1)^{a} \ 1 & 0 \end{array}\right]$and$S_{a}=\frac{1}{2}\left[ \begin{array}{lr} L_{k,a} & \Delta_{a} \ 1 & L_{k,a} \end{array}\right]$, where ${\mathrm{\Delta }}_a=L_{k,a}^2-4(-1{)}^a$. The most notable side of this paper is our proof method, since all the identities used in the proofs of main theorems are proved previously by using the matrices $R_a$ and $S_a$, with $a$ an natural number. Although the identities we proved are known, our proofs are not encountered in the $k$-Fibonacci and $k$-Lucas numbers literature.