Value expressions

Question

What is the value of the expression: $\sum_{k=0}^{d}b^{k}\left[\begin{array}{c} d\\ k \end{array}\right]\prod_{i=0}^{k-1}\left(c-b^{i}\right) $ for $ k = 0 $?

In particular, what is then the value of: $\prod_{i=0}^{-1}\left(c-b^{i}\right)$?

Answer 1

Its same as multiplying no numbers together so the series $$\prod_{i=0}^{-1}\left(c-b^{i}\right)=1$$ in fact this works for all negative $n$ so that $$\prod_{i=0}^{-n}\left(c-b^{i}\right)=1$$ because her too we aren't multiplying any numbers!!