# Is this recurrent sequence decreasing?

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.