1760887 I've been working on this homework problem for a while now and can't seem to solve it. Let $L_n = L_{n-1} + L_{n-2}$ for $n\ge 2$ where $L_0 = 2$ and $L_1 = 1$
$M_n = 1 + \sum_{i=0}^n{L_i}$ Calculate the first few elements of the sequence $M_n$ to observe a simpler formula relating it to the sequence $L_n$. Prove your conjectural formula by using closed expressions for the generating functions $L(z)$ and $M(z)$ of the sequences $L_n$ and $M_n$.
now I've found $M_n = L_{n+2}$
Defined the closed formula of the generating function of a lucas sequence as $L(z)=\left(\frac{1}{1- \frac{1+\sqrt5}2z}\right)+\left(\frac{1}{1- \frac{1-\sqrt5}2z}\right)$ but now every time I try to solve $M_n$ I end up with generating functions defined by finite series and I don't know how to find the closed formula of those, any help would be appreciated.
HINT: Take advantage of the convolution hiding in the definition of $M_n$. Let
$$f(z)=\frac1{1-z}=\sum_{n\ge 0}z^n\;;$$
then
$$L(z)f(z)=\sum_{n\ge 0}\left(\sum_{k=0}^nL_k\cdot 1\right)z^n=\sum_{n\ge 0}(M_n-1)z^n=\sum_{n\ge 0}M_nz^n-\sum_{n\ge 0}z^n\;,$$
so
$$\sum_{n\ge 0}M_nz^n=L(z)f(z)+\sum_{n\ge 0}z^n\;.$$