HMMT 2025 U17

Let $f$ be a quadratic polynomial with real coefficients, and let $g_{1}$ , $g_{2}$ , $g_{3}$ , ... be a geometric progression of real numbers. Define $a_{n} = f(n) + g_{n}$ . Given that $a_{1}$ , $a_{2}$ , $a_{3}$ , $a_{4}$ , and $a_{5}$ are equal to 1, 2, 3, 14, and 16, respectively, compute $\frac{g_{2}}{g_{1}}$ .