Solution: The first 5 terms are $ a_1 = 5 $, $ a_2 = 8 $, $ a_3 = 11 $, $ a_4 = 14 $, and $ a_5 = 17 $. Their sum is $ 5 + 8 + 11 + 14 + 17 = 55 $. Dividing 55 by 7 gives a quotient of 7 and a remainder of $ 55 - 7 \times 7 = 55 - 49 = 6 $.

["Understanding Modular Arithmetic: The Step-by-Step Breakdown of a Remainder Calculation", "In mathematics, understanding how numbers behave under division is key to solving problems involving remainders, divisibility, and patterns. A simple yet insightful example involves computing the sum of an arithmetic sequence and then determining the remainder when this sum is divided by a specific divisor—in this case, 7.", "Let’s explore the solution step by step, highlighting the core concept and broader relevance.", "### The Arithmetic Sequence", "We begin with the first five terms of an arithmetic sequence defined as:\n- $ a_1 = 5 $\n- $ a_2 = 8 $\n- $ a_3 = 11 $\n- $ a_4 = 14 $\n- $ a_5 = 17 $", "Each term increases by 3:\n$ a_2 - a_1 = 3 $, $ a_3 - a_2 = 3 $, and so on.", "### Step 1: Calculating the Sum", "We now sum these five values:", "$$\na_1 + a_2 + a_3 + a_4 + a_5 = 5 + 8 + 11 + 14 + 17\n$$", "$$\n= 5 + 8 = 13 \\n13 + 11 = 24 \\n24 + 14 = 38 \\n38 + 17 = 55\n$$", "Thus, the total sum is 55.", "### Step 2: Dividing by 7 – Division with Quotient and Remainder", "We divide 55 by 7 to investigate the remainder:", "$$\n55 \div 7 = ?\n$$", "Performing the division:\n- $ 7 \ imes 7 = 49 $, which is the largest multiple of 7 less than or equal to 55\n- So, the quotient is 7", "The remainder is found by subtracting:", "$$\n55 - (7 \ imes 7) = 55 - 49 = 6\n$$", "### Summary", "- The sum of the five terms is 55\n- Dividing 55 by 7 yields:", "$$\n55 = 7 \ imes 7 + 6\n$$", "Where 7 is the quotient and 6 is the remainder.", "### Why This Matters: Modular Arithmetic in Real Life", "This simple process introduces modular arithmetic, where numbers are evaluated based on their remainder after division by a fixed number (here, 7). Such concepts are foundational in cryptography, computer science, coding theory, and daily problem-solving involving patterns and cycles.", "Understanding how to break down numbers this way helps students and learners alike develop strong number sense and algebraic intuition.", "---", "Key Takeaway:\nStarting from a clear arithmetic sequence, computing the sum, then dividing to find quotient and remainder allows deeper insight into modular relationships—an essential skill in both academic math and real-world applications.", "Whether you’re solving homework problems or building algorithms, mastering these steps strengthens your foundation in mathematics."]