Question

Please show this equality.

n-1 p) p

Answer 1

$\sum_{k=1}^{n-1}(1-p)^{k}p^{n-k}=p^{n-1}(1-p)^{1}+p^{n-2}(1-p)^{2}+p^{n-3}(1-p)^{3}+.....+p^{1}(1-p)^{n-1$

$=p(1-p)[p^{n-2}+p^{n-3}(1-p)+p^{n-4}(1-p)^{2}+.....+(1-p)^{n-2}]$

$=p(1-p)[p^{n-2}\frac{(\frac{1-p}{p})^{n-1}-1}{\frac{1-p}{p}-1}]=p(1-p)[p^{n-2}.\frac{p^{n-1}-(1-p)^{n-1}}{p-(1-p)}]$

$=p^{n-1}(1-p)[\frac{p^{n-1}-(1-p)^{n-1}}{p-(1-p)}]$

which is the correct equality.