The probability is $\boxed{0.2458}$.Question: What is the remainder when the sum of the first 5 terms of the sequence $ a_n = 3n + 2 $ is divided by 7?

["Title: Remainder When the Sum of First 5 Terms of $ a_n = 3n + 2 $ Is Divided by 7", "---", "Introduction\nWhen working with sequences in mathematics, one common task is to compute the sum of the first few terms and determine the remainder after division by a given number—often modulo arithmetic. In this article, we analyze the sequence defined by $ a_n = 3n + 2 $, compute the sum of its first 5 terms, and find the remainder when this sum is divided by 7. The key probability value $\boxed{0.2458}$ referenced here is not directly tied to the modular arithmetic, but the sequence’s structure and sum behavior still offer rich insight into arithmetic patterns.", "---", "Step 1: Understand the Sequence\nThe sequence is defined by the formula:\n$$\na_n = 3n + 2\n$$\nThis generates the following values for $ n = 1 $ to $ 5 $:\n- $ a_1 = 3(1) + 2 = 5 $\n- $ a_2 = 3(2) + 2 = 8 $\n- $ a_3 = 3(3) + 2 = 11 $\n- $ a_4 = 3(4) + 2 = 14 $\n- $ a_5 = 3(5) + 2 = 17 $", "---", "Step 2: Compute the Sum of the First 5 Terms\nNow, add these first 5 terms:\n$$\nS = 5 + 8 + 11 + 14 + 17\n$$\nGrouping pairs simplifies the calculation:\n$$\n(5 + 17) + (8 + 14) + 11 = 22 + 22 + 11 = 55\n$$\nSo, the sum $ S = 55 $", "---", "Step 3: Find the Remainder When Divided by 7\nWe now compute $ 55 \mod 7 $:\nDivide 55 by 7:\n$$\n7 \ imes 7 = 49 \quad \ ext{and} \quad 55 - 49 = 6\n$$\nTherefore,\n$$\n55 \div 7 = 7 \ ext{ remainder } 6 \quad \Rightarrow \quad 55 \equiv 6 \pmod{7}\n$$", "---", "Conclusion\nThe remainder when the sum of the first 5 terms of the sequence $ a_n = 3n + 2 $ is divided by 7 is $ \boxed{6} $.\nWhile the probability $\boxed{0.2458}$ appears earlier in the problem context, it does not affect the direct modular calculation. However, the structured nature of arithmetic sequences ensures predictable sums—ideal for modular analysis and algorithmic thinking.", "---", "Bonus Insight: Connection to the Given Probability\nIf $\boxed{0.2458}$ represents a probability derived from similar counting or ratio-based models (e.g., favorable outcomes over total outcomes), here the sequence sum modulo 7 offers a deterministic remainder rather than a probabilistic outcome—highlighting how deterministic number patterns inform both algebra and discrete probability.", "---", "Keywords:\nsequence sum, modular arithmetic, remainder mod 7, arithmetic sequence $ a_n = 3n + 2 $, sum of first 5 terms, modular remainder, $ 55 \mod 7 $, 0.2458 probability context", "---", "Meta Description:\nCalculate the sum of the first 5 terms of $ a_n = 3n + 2 $, find the remainder when divided by 7, and understand modular arithmetic in sequences—with full steps and connection to probabilistic thinking."]