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:

I tried to prove this by taking the difference between consecutive terms of the summand i.e. $(px_t^2 – (p+q)x_t) – (px_{t+1}^2 – (p+q)x_{t+1}))$ and trying to show that it is greater than 0 but I am not able to make progress on this.