$$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?

["Title: Finding the Smallest Integer $ r $ for Which the Sum of a Geometric Sequence Exceeds 300", "When modeling time series data, statisticians often rely on geometric sequences to capture exponential growth or decay. A classic example involves a time series defined by a geometric sequence with first term $ a = 3 $ and common ratio $ r $. Understanding when the sum of the first five terms exceeds 300 helps determine thresholds in predictive models.", "In a geometric sequence, the sum of the first $ n $ terms is given by the formula:", "$$\nS_n = a \frac{r^n - 1}{r - 1} \quad \ ext{(for } r <br/>\neq 1\ ext{)}\n$$", "Here, $ a = 3 $, $ n = 5 $, so:", "$$\nS_5 = 3 \cdot \frac{r^5 - 1}{r - 1}\n$$", "We want the smallest integer $ r \geq 2 $ (since $ r = 1 $ gives a linear sum and fails the condition) such that:", "$$\n3 \cdot \frac{r^5 - 1}{r - 1} > 300\n$$", "Divide both sides by 3:", "$$\n\frac{r^5 - 1}{r - 1} > 100\n$$", "Note that $ \frac{r^5 - 1}{r - 1} $ is the sum of a geometric series with 5 terms starting at 1, so it equals $ 1 + r + r^2 + r^3 + r^4 $. But since our sequence starts at 3, the formula applied earlier remains correct: $ S_5 = 3(1 + r + r^2 + r^3 + r^4) $. So we can equivalently analyze:", "$$\n3(1 + r + r^2 + r^3 + r^4) > 300\n\quad \Rightarrow \quad\n1 + r + r^2 + r^3 + r^4 > 100\n$$", "Now test successive integer values of $ r $:", "- For $ r = 3 $:\n $ 1 + 3 + 9 + 27 + 81 = 121 > 100 $ ✅\n- For $ r = 2 $:\n $ 1 + 2 + 4 + 8 + 16 = 31 < 100 $ ❌", "So $ r = 3 $ satisfies the inequality, but $ r = 2 $ does not. Since $ r = 3 $ is an integer and the smallest such value (checking $ r = 3 $ is the first integer where the sum exceeds 100), it is the answer.", "Conclusion: The smallest integer value of $ r $ for which the sum of the first five terms exceeds 300 is $ r = 3 $. This threshold helps statisticians determine when exponential growth dynamics become significant in time series forecasting.", "---", "Keywords: geometric sequence, time series modeling, sum of geometric series, integer value of $ r $, exponential growth, statistical modeling, $ r^5 > 100 $ condition, smallest integer $ r $, cumulative sum formula", "Meta Description:\nDiscover the smallest integer common ratio $ r $ in a geometric time series with first term 3 such that the sum of the first five terms exceeds 300. Learn how $ 3(1 + r + r^2 + r^3 + r^4) > 300 $ determines exponential model thresholds."]