When solving a problem in Markov chains, I was faced with the task of proving if the following inequality is true or not $$ 1 - a\cdot f_k - \sum_{i=1}^n f_kp^i \leq (1-af_kp)^n , \ \forall n\in \mathbb N $$
Note that $a\in \mathbb N$ and $f_k, p \in (0,1)$. Can anyone help me check if this inequality is true or not? Bonus: if the inequality is not true, is there a $\beta \in (0,1)$ for which
$$ 1 - a\cdot f_k - \sum_{i=1}^n f_kp^i \leq \beta ^n , \ \forall n\in \mathbb N $$
As one of many counterexamples, if $$ \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! n=2,\;\;\; a=3,\;\;\; p=\frac{3}{4},\;\;\; f_k=\frac{1}{28} $$ then$\;\text{LHS}-\text{RHS}={\Large{\frac{3}{12544}}} > 0$.
As regards the bonus question, for fixed $a,p$, choose $f_k\in(0,1)$ such that $$ f_k < \min \Bigl( \frac{1}{2a}, \frac{1-p}{2p} \Bigr) $$ Then we have $$ af_k < \frac{1}{2}\;\;\;\text{and}\;\;\;f_k\Bigl(\frac{p}{1-p}\Bigr) < \frac{1}{2} $$ hence there is no $\beta\in(0,1)$ such that $$ 1-af_k-\sum_{i=1}^n f_kp^i \le \beta^n $$ for all $n$, since as $n$ approaches infinity the $\text{LHS}$ approaches $$ 1-af_k-f_k\Bigl(\frac{p}{1-p}\Bigr) > 1-\frac{1}{2}-\frac{1}{2} = 0 $$ whereas $\beta^n$ approaches $0$.