  A positive linear recurrence sequence (PLRS) is a sequence defined by a
homogeneous linear recurrence relation with positive coefficients and a
particular set of initial conditions. A sequence of positive integers is
\emph{complete} if every positive integer is a sum of distinct terms of the
sequence. One consequence of Zeckendorf's theorem is that the sequence of
Fibonacci numbers is complete. Previous work has established a generalized
Zeckendorf's theorem for all PLRS's. We consider PLRS's and want to classify
them as complete or not. We study how completeness is affected by modifying the
recurrence coefficients of a PLRS. Then, we determine in many cases which
sequences generated by coefficients of the forms $[1, \ldots, 1, 0, \ldots, 0,
N]$ are complete. Further, we conjecture bounds for other maximal last
coefficients in complete sequences in other families of PLRS's. Our primary
method is applying Brown's criterion, which says that an increasing sequence
$\{H_n\}_{n = 1}^{\infty}$ is complete if and only if $H_1 = 1$ and $H_{n + 1}
\leq 1 + \sum_{i = 1}^n H_i$. This paper is an introduction to the topic that
is explored further in Completeness of Positive Linear Recurrence Sequences
arXiv:2010.01655.
