$(Y_n)_n$ is a sequence of random variable i.i.d such that $Y_1=1$ with probability $p$ and $Y_1=-k$ with probability $1-p$.
$(S_n)_n$ is a sequence of random variables defined as $S_0=0$ and $S_n=\sum_{i=0}^n Y_i$.
I have to show that if $\rho=P(\exists n \in \mathbb{N} : S_n\geq 1)$ then $\rho=p+(1-p)\rho^{k+1}$.
I don't really know how to start so any help is appreciate.
Start by conditioning on the first step:
\begin{eqnarray*} \rho &=& P(Y_1=1)P(S_n\geq 1\mid Y_1=1) + P(Y_1=-k)P(S_n\geq 1\mid Y_1=-k) \\ &=& p\cdot 1 + (1-p)P(S_n \geq k+1) \\ && \qquad\text{since, if $Y_1=-k,\;$ subsequently reaching $S_n=1,\;$ for some $n,\;$ is} \\ && \qquad\text{equivalent to reaching $S_m=k+1,\;$ starting at $0,\;$ for some $m$.} \end{eqnarray*}
Now, any path from $0$ to $k+1$ consists of $k+1$ contiguous paths that each "go up" by $1$. And, conversely, any $k+1$ contiguous such paths form a path from $0$ to $k+1$. So, $P(S_n\geq k+1) = P(S_1\geq 1)^{k+1} = \rho^{k+1}$, hence the result.