I want to confirm the authenticity of this statement. I disagree with this because sum of any AP can be represented as $$\text{Sum} = \frac{n^2d}{2} + an - \frac{nd}{2},$$ where $n$ is total number of terms, $d$ is common difference, and $a$ is first term of an AP.
Now when I compare the above equation with general form of a quadratic equation i.e $px^2 + qx + c = 0$, I get $p = \frac{d}{2}$ and $q = a - \frac{d}{2}$ (taking the Sum of an Arithmetic Progression as a quadratic equation in $n$) But where is $c$?
The correct statement in my opinion would be:
Any equation of the form $px^2 + qx = 0$ is the sum of some AP.
Firstly, as it is stated, is not true, as $x$ "used as" $n$ needs to be a whole number, and not all equations produce whole numbers, so the original statement might not say "equation", rather "expression" or "function (of x)", and $x\in\mathbb{Z}$, $x > 1$ should be set as conditions.
The two things, that $px^2 + qx$ is an AP and $px^2 + qx + c$ is an AP are not mutually exclusive, on the contrary: you did well to show that $px^2 + qx$ is an AP. As $n$ is the number of terms, if you add $\frac{c}{n}$ to each term, what do you get? Is that an AP? What is its sum?
Regarding the $Sum$ formula for $px^2 + qx$, how does it change if you use $a' = a + \frac{c}{n}$ for $px^2 + qx + c$, while $d$ and $n$ are unchanged?