What is it to solve an equation forward? I'm reading a book in Monetary Economics and I don't understand a step.
I have this expression:
$$ \dfrac{\lambda_{t}}{P_{t}} = \beta \left( \dfrac{\lambda_{t+1} + \mu_{t+1}}{P_{t+1}} \right) $$
And then it says "solving this equation forward implies that":
$$ \dfrac{\lambda_{t}}{P_{t}} = \sum_{i=1}^{\infty} \beta^{i} \left( \dfrac{\mu_{t+i}}{P_{t+i}} \right) $$
I dont't know what they're doing. What is it to solve an equation forward?

We can begin to solve this equation in two ways, an idea that generalizes to every time we have a relationship between the $t^{\text{th}}$ and $(t+1)^{\text{th}}$ term of a sequence.
One way is to start with the equation for $\frac{\lambda_{t}}{P_{t}}$, and use substitution to eliminate $\frac{\lambda_{t+1}}{P_{t+1}}$, then $\frac{\lambda_{t+2}}{P_{t+2}}$, and so on:
\begin{align}
\frac{\lambda_t}{P_t} &= \beta \frac{\mu_{t+1}}{P_{t+1}} + \beta \frac{\lambda_{t+1}}{P_{t+1}} \\
  &= \beta \frac{\mu_{t+1}}{P_{t+1}} + \beta \left(\beta \frac{\mu_{t+2}}{P_{t+2}} + \beta \frac{\lambda_{t+2}}{P_{t+2}}\right)  \\
  &=  \beta \frac{\mu_{t+1}}{P_{t+1}} + \beta^2 \frac{\mu_{t+2}}{P_{t+2}} + \beta^2 \frac{\lambda_{t+2}}{P_{t+2}} \\
  &= \beta \frac{\mu_{t+1}}{P_{t+1}} + \beta^2 \frac{\mu_{t+2}}{P_{t+2}} + \beta^2 \left(\beta \frac{\mu_{t+3}}{P_{t+3}} + \beta \frac{\lambda_{t+3}}{P_{t+3}}\right) \\
  &= \beta \frac{\mu_{t+1}}{P_{t+1}} + \beta^2 \frac{\mu_{t+2}}{P_{t+2}} + \beta^3 \frac{\mu_{t+3}}{P_{t+3}} + \beta^3 \frac{\lambda_{t+3}}{P_{t+3}} \\
 &= \dots
\end{align}
The other way is to rewrite this equation as $\frac{\lambda_{t+1}}{P_{t+1}} = \beta^{-1} \frac{\lambda_t}{P_t} - \frac{\mu_{t+1}}{P_{t+1}}$ or better yet $\frac{\lambda_t}{P_t} = \beta^{-1} \frac{\lambda_{t-1}}{P_{t-1}} - \frac{\mu_t}{P_t}$, and then use substitution to eliminate $\frac{\lambda_{t-1}}{P_{t-1}}$, then $\frac{\lambda_{t-2}}{P_{t-2}}$, and so on:
\begin{align}
\frac{\lambda_t}{P_t} &= \beta^{-1} \frac{\lambda_{t-1}}{P_{t-1}} - \frac{\mu_t}{P_t} \\
  &= \beta^{-1} \left(\beta^{-1} \frac{\lambda_{t-2}}{P_{t-2}} - \frac{\mu_{t-1}}{P_{t-1}}\right)- \frac{\mu_t}{P_t} \\
  &= \beta^{-2} \frac{\lambda_{t-2}}{P_{t-2}} - \beta^{-1}\frac{\mu_{t-1}}{P_{t-1}} - \frac{\mu_t}{P_t} \\
  &= \beta^{-2} \left(\beta^{-1} \frac{\lambda_{t-3}}{P_{t-3}} - \frac{\mu_{t-2}}{P_{t-2}}\right)- \beta^{-1}\frac{\mu_{t-1}}{P_{t-1}} - \frac{\mu_t}{P_t} \\
  &= \beta^{-3} \frac{\lambda_{t-3}}{P_{t-3}} - \beta^{-2}\frac{\mu_{t-2}}{P_{t-2}}- \beta^{-1}\frac{\mu_{t-1}}{P_{t-1}} - \frac{\mu_t}{P_t} \\
  &= \dots
\end{align}
I feel like it makes sense to say that in the first method, we're solving the equation "forward", since we're getting higher and higher indices on $\lambda$. In the second method, we're solving the equation "backward", getting lower and lower indices.
In the first method, assuming that $\beta^i \frac{\lambda_{t+i}}{P_{t+i}} \to 0$ as $i \to \infty$, we're eventually left with just an infinite sum with no $\lambda$'s in it. In the second method, we're left with a finite sum with no $\lambda$'s in it, assuming that we have an initial condition for $\frac{\lambda_0}{P_0}$.
