HMMT 二月 2013 · 代数 · 第 2 题

HMMT February 2013 — Algebra — Problem 2

Let { a } be an arithmetic sequence and { g } be a geometric sequence such that the first four n n ≥ 1 n n ≥ 1 terms of { a + g } are 0, 0, 1, and 0, in that order. What is the 10th term of { a + g } ? n n n n x y z

解析

Let { a } be an arithmetic sequence and { g } be a geometric sequence such that the first four n n ≥ 1 n n ≥ 1 terms of { a + g } are 0, 0, 1, and 0, in that order. What is the 10th term of { a + g } ? n n n n 2 3 Answer: − 54 Let the terms of the geometric sequence be a, ra, r a, r a . Then, the terms of 2 3 the arithmetic sequence are − a, − ra, − r a + 1 , − r a . However, if the first two terms of this sequence are − a, − ra , the next two terms must also be ( − 2 r + 1) a, ( − 3 r + 2) a . It is clear that a 6 = 0 because 3 a + g 6 = 0, so − r = − 3 r +2 ⇒ r = 1 or − 2. However, we see from the arithmetic sequence that r = 1 3 3 is impossible, so r = − 2. Finally, by considering a , we see that − 4 a + 1 = 5 a , so a = 1 / 9. We also 3 n − 1 see that a = (3 n − 4) a and g = ( − 2) a , so our answer is a + g = (26 − 512) a = − 486 a = − 54. n n 10 10 x y z