I am confused to understand and to find maximal $\ k=k(n,p)$ and minimal $\ k=k(n,p)$ for a binomial distribution question. Below are my question:
Q1. Let $\Pr=binomial(n,p)$ . Check that $\Pr(i+1)=\frac{p}{1−p}⋅\frac{n−i}{i+1}⋅Pr(i)$ for all $\ i=0,1,…,n−1$ and find the maximal $\ k=k(n,p)$, such that $\ Pr(0),…,Pr(k)$ is a strictly increasing sequence: $\ k(n,p)=$ (answer to be filled in)
Q2. Let $\Pr=binomial(n,p)$. Find the minimal $\ k=k(n,p)$, such that $\Pr(k),Pr(k+1),…$ is a strictly decreasing sequence: $\ k(n,p)=$ (answer to be filled in)
To understand these two questions, I try to find a clue from Wikipedia binomial distribution. The closest thing that I found is monotone increasing and monotone decreasing under Probability mass function section. However, it seems like I cannot apply it directly. Therefore, I would like to ask that did I understand this question correctly?