HMMT 二月 2013 · 代数 · 第 2 题

英文解析

2. Let { a } be an arithmetic sequence and { g } be a geometric sequence such that the first fourn n ≥ 1 n n ≥ 1
   terms of { a + g } are 0, 0, 1, and 0, in that order. What is the 10 th 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 sequenceare − a, − ra , the next two terms must also be ( − 2 r + 1) a, ( − 3 r + 2) a . It is clear that a 6 = 0 becausea + g 6 = 0, so − r = − 3 r +2 ⇒ r = 1 or − 2. However, we see from the arithmetic sequence that r = 13
   3 3
   is impossible, so r = − 2. Finally, by considering a , we see that − 4 a + 1 = 5 a , so a = 1 / 9. We alson − 13
   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