  In this work we consider extensions of solvable Lie algebras with naturally
graded filiform nilradicals. Note that there exist two naturally graded
filiform Lie algebras $n_{n, 1}$ and $Q_{2n}.$ We find all one-dimensional
central extensions of the algebra $n_{n, 1}$ and show that any extension of
$Q_{2n}$ is split. After that we find one-dimensional extensions of solvable
Lie algebras with nilradical $n_{n, 1}$. We prove that there exists a unique
non-split central extension of solvable Lie algebras with nilradical $n_{n, 1}$
of maximal codimension. Moreover, all one-dimensional extensions of solvable
Lie algebras with nilradical $n_{n, 1}$ whose codimension is equal to one are
found and compared these solvable algebras with the solvable algebras with
nilradicals are one-dimensional central extension of algebra $n_{n, 1}$.
