Thank you for reporting, we will resolve it shortly
Q.
If $\frac{^{n}P_{r-1}}{a} = \frac{^{n}P_{r}}{b} = \frac{^{n}P_{r+1}}{c}$, then
Permutations and Combinations
Solution:
$\frac{1}{a} \frac{n!}{\left(n - r + 1\right)!} $
$= \frac{1}{b} \frac{n!}{\left(n-r\right)!} $
$= \frac{1}{c} \frac{n!}{\left(n - r - 1\right)! }$
$ \Rightarrow \frac{b}{a} = n - r + 1 \quad....\left(i\right)$ and
$ \frac{c}{b} = n - r \quad...\left(ii\right)$
From $\left(i\right)$ and $\left(ii\right)$, we get
$ \frac{b}{a} = \frac{c}{b} +1 $
$ \Rightarrow b^{2} = a\left(b+c\right)$.