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Home » The least common multiple is $ \boxed2160 $. - Dyverse

The least common multiple is $ \boxed2160 $. - Dyverse

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 $$Question: A statistician models a time series using a geometric sequence where the first term is $ 3 $ and the common ratio is $ r $. If the sum of the first five terms exceeds $ 300 $, what is the smallest integer value of $ r $ that satisfies this condition?

📌 The sum of the first five terms of a geometric sequence with first term $ a = 3 $ and common ratio $ r $ is given by:

📌 S_5 = 3(1 + r + r^2 + r^3 + r^4) = 3 \cdot rac{r^5 - 1}{r - 1} \quad ext{(for } r

📌 e 1 ext{)}

📌 \cdot rac{r^5 - 1}{r - 1} > 300

📌 rac{r^5 - 1}{r - 1} > 100

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

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