I'm given the first term $a$ and the common term $d$. I need to find an index $n$, so that the sum of first terms is $S(n)=v$.
Basically, I need to solve the following quadratic system in terms of $n$: $$\frac{2a+d(n-1)}{2}n=v$$
How to prove or give a counterexample to the following proposition: there is at most one positive integer root of the equation $\frac{2a+d(n-1)}{2}n=v$.
A counterexample to the statement given would be where $d\lt 0$. For example, if $a=1$ and $d=-1$, then the values of $S(n)$ would be $$1,1,0,-2,...$$ where the initial two values are the same. So, $S(n)=1$ would have $2$ positive solutions - $n=1,2$.