Question 1
Let $\alpha$ be the positive root of $x^2+x=5$. Suppose that $n$ is a positive integer and that nonnegative integers $c_0,c_1,\ldots,c_n$ satisfy
$$c_0+c_1\alpha+c_2\alpha^2+\cdots+c_n\alpha^n=2015.$$
(a) Prove that $c_0+c_1+\cdots+c_n\equiv2\pmod3$.
(b) Find the least possible value of $c_0+c_1+\cdots+c_n$.
$$c_0+c_1\alpha+c_2\alpha^2+\cdots+c_n\alpha^n=2015.$$
(a) Prove that $c_0+c_1+\cdots+c_n\equiv2\pmod3$.
(b) Find the least possible value of $c_0+c_1+\cdots+c_n$.