Let $S_n$ be defined as $\frac{1}{n}\sum_{t=1}^{t=n} [px_t^2 - (p+q)x_t]$ where $x_t = 1-(1-p-q)^t$. We want to find the conditions on $p$ and $q$ such that $S_n$ is monotonically decreasing for all $n$. $0 < p,q < 1$ and $-1 < 1-p-q < 1$.

Note: Till now I have tried to get a closed bound expression for $S_n$ and differentiate it w.r.t. $n$ to get the conditions for a negative slope but it is getting really complex.

This is similar to a previous question of mine but by mistake, I wrote the wrong polynomial there.