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.