I’m confused why this infinite geometric sum is true m

Question

I’m really confused why this is true $$p \cdot \sum\limits_{i=m+1}^\infty (1-p)^{i-1} = (1-p)^m$$ I know the formula for the infinite geometric sum is $$\sum\limits_{i=0}^\infty a r^i = \frac{a}{1-r}$$

Answer 1

Read the formula for the sum of a geometric series as $$\frac{\text{first term}}{\text{$1-$ common ratio}}.$$ Here the first term is obtained by plugging $i=m+1$: $$p(1-p)^m$$ and the common ratio is: $$1-p$$