\(a_5 = 3(25) + 10 + 1 = 75 + 10 + 1 = 86\).

["Decoding (a_5 = 3(25) + 10 + 1): A Clear Breakdown of the Calculation", "Mathematics is filled with elegant expressions that combine operations in clever ways to deliver surprising results. One such expression is (a_5 = 3(25) + 10 + 1). While it might initially seem like a routine arithmetic problem, understanding how this evaluation unfolds reveals key mathematical concepts like order of operations, distributive property, and basic algebraic thinking.", "### What Does (a_5 = 3(25) + 10 + 1) Mean?", "In contexts involving sequences or recursive definitions, (a_n) often represents a term in a sequence or a formula designed to compute values based on (n). Here, the formula defines:", "[\na_5 = 3 \ imes 25 + 10 + 1\n]", "But to fully appreciate this expression, we break it down step by step using proper order of operations.", "### Step-by-Step Evaluation", "1. Parentheses First (Distribution):\n The term (3(25)) must be evaluated first due to the multiplication inside parentheses.\n [\n 3 \ imes 25 = 75\n ]", "2. Addition Next:\n Now substitute back and proceed with left-to-right addition:\n [\n 75 + 10 + 1\n ]\n - (75 + 10 = 85)\n - (85 + 1 = 86)", "So,\n[\na_5 = 3(25) + 10 + 1 = 86\n]", "### Why This Calculation Matters", "At first glance, the expression appears straightforward, but mastering each step builds a foundation for solving more complex algebraic equations. It reinforces the importance of:", "- Order of Operations (PEMDAS/BODMAS): Parentheses → Exponents → Multiplication/Division → Addition/Subtraction. Missing any step disrupts the correct result.\n- Distributive Property: Recognizing that multiplication applies across parentheses ensures accurate expansion.\n- Algebraic Evaluation: Even in simplified number expressions, understanding function evaluation ((a_n) meaning the value at position (n)) prepares learners for variable-based sequences.", "### Real-World and Educational Applications", "Expressions like (a_5 = 3 \ imes 25 + 10 + 1) aren’t just abstract exercises—they model real-world situations such as calculating total cost (e.g., 3 sets of $25, 10 additional fees, and 1 tax), or constructing algorithmic logic in programming.", "In classrooms, solidifying these computational steps helps students transition smoothly from basic arithmetic to algebraic reasoning.", "### Conclusion", "While (a_5 = 3(25) + 10 + 1 = 86) appears simple, unpacking it highlights fundamental mathematical habits: careful order of operations, precise arithmetic, and clear function evaluation. Whether solving equations, coding logic, or everyday calculations, mastering these foundational steps builds confidence and competence.", "Understanding how every part of the expression contributes to the final number empowers you to tackle increasingly complex problems with clarity and accuracy.", "---", "Keywords:\nmath calculation (a_5 = 3(25) + 10 + 1), order of operations, distributive property, algebraic evaluation, step-by-step math, arithmetic basics, solving equations, computational thinking."]