HMMT 十一月 2023 · THM 赛 · 第 1 题

HMMT November 2023 — THM Round — Problem 1

Tyler has an infinite geometric series with sum 10 . He increases the first term of his sequence by 4 and swiftly changes the subsequent terms so that the common ratio remains the same, creating a new geometric series with sum 15 . Compute the common ratio of Tyler’s series.

解析

Tyler has an infinite geometric series with sum 10. He increases the first term of his sequence by 4 and swiftly changes the subsequent terms so that the common ratio remains the same, creating a new geometric series with sum 15. Compute the common ratio of Tyler’s series. Proposed by: Isabella Quan 1 Answer: 5 Solution: Let a and r be the first term and common ratio of the original series, respectively. Then a a +4 = 10 and = 15. Dividing these equations, we get that 1 − r 1 − r a + 4 15 = = ⇒ a = 8 . a 10 a 8 1 Solving for r with = = 10 gives r = . 1 − r 1 − r 5