A positive integer $n$ is such that $1-2x+3x^2-4x^3+5x^4-...-2014x^{2013}+nx^{2014}$ has at least one integer solution. Find $n$.

Question

A positive integer $n$ is such that $$1-2x+3x^2-4x^3+5x^4-...-2014x^{2013}+nx^{2014}$$ has at least one integer solution. Find $n$.

Answer 1

By the rational root test, the possible rational roots of the polynomial have the form $$ r=\frac{1}{k} $$ where $k$ is an integer (positive or negative) that divides $n$. Which of them are integer? When are they roots of the polynomial?

Answer 2

If you believe the question, that you are looking for one possible value of $n$, then you should be able to find one by setting $f(1)=0$ which clearly gives you a linear equation for $n$. Why doesn't $f(-1)=0$ work?

You need something like @egreg's approach to show there are no other possibilities.