what will be the $(t+1)$-th term in this summation expansion and why?

Question

What will be the last term in the following summation $$\sum_{k=0}^{t}F_{t}F_{t-1}\cdots F_{k+1}x_k.$$

Answer 1

This is a summation of finitely many terms; talking about "last term" is a bit ambiguous. However, in the order that you've specified, we may find the "last term" by rewriting as $$ \sum _{k=0}^{t} F_t F_{t-1} \cdots F_{k+1}x_k = \sum _{k=0}^{t} \left( x_k \prod _{j=k+1}^{t} F_j \right). $$ We can see that $$ \prod _{j=k+1}^{t} F_j $$ is undefined for $k=t$. This leaves two possibilities:

1. The author has some convention regarding this situations where the product is taken as $1$, in which case the last term is $x_t$
2. There is a mistake in the summation limit and $k$ is supposed to reach $t-1$, in which case the last term is $x_{t-1}F_t$