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Home » A geometric sequence has first term 4 and fifth term 1024. What is the common ratio? - ECD Germany

A geometric sequence has first term 4 and fifth term 1024. What is the common ratio? - ECD Germany

📅 May 9, 2026 👤 scraface

May 09, 2026

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📌 rac{r^5 - 1}{r - 1} > 100

📌 We test integer values of $ r > 1 $ (since $ r = 1 $ gives $ S_5 = 15 $, too small):

📌 For $ r = 2 $:

📌 rac{32 - 1}{2 - 1} = 31 \quad \Rightarrow \quad 3 \cdot 31 = 93 < 300

📌 For $ r = 3 $:

📌 rac{243 - 1}{3 - 1} = rac{242}{2} = 121 \quad \Rightarrow \quad 3 \cdot 121 = 363 > 300

📌 Thus, $ r = 3 $ satisfies the condition, and since $ r = 2 $ does not, the smallest integer $ r $ is $ oxed{3} $.

📌 Question: A primatologist observes that the number of vocalizations a primate group makes each hour follows the recurrence $ v_n = 2v_{n-1} - 7 $, with $ v_1 = 10 $. What is the value of $ v_5 $?

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