HMMT 二月 2025 · 冲刺赛 · 第 17 题

HMMT February 2025 — Guts Round — Problem 17

[11] Let f be a quadratic polynomial with real coefficients, and let g , g , g , . . . be a geometric progression 1 2 3 of real numbers. Define a = f ( n ) + g . Given that a , a , a , a , and a are equal to 1, 2, 3, 14, and n n 1 2 3 4 5 g 2 16, respectively, compute . g 1

解析

[11] Let f be a quadratic polynomial with real coefficients, and let g , g , g , . . . be a geometric 1 2 3 progression of real numbers. Define a = f ( n ) + g . Given that a , a , a , a , and a are equal to 1, n n 1 2 3 4 5 g 2 2, 3, 14, and 16, respectively, compute . g 1 Proposed by: Pitchayut Saengrungkongka 19 Answer: − 10 Solution: We will use the method of finite differences. Define b = a − 3 a + 3 a − a . Since n n +3 n +2 n +1 n f is quadratic, the third finite difference of f is zero. So, b = g − 3 g + 3 g − g . Letting n n +3 n +2 n +1 n 3 2 the common ratio of the geometric sequence be r , we get that b = ( r − 3 r + 3 r − 1) g . So, b is n n n g b 2 2 a constant multiple of g . Thus the ratio = . Computing b = 14 − 3 · 3 + 3 · 2 − 1 = 10 and n 1 g b 1 1 b = 16 − 3 · 14 + 3 · 3 − 3 · 2 = − 19, we get 2 g b 19 2 2 = = − . g b 10 1 1