\(a_{10} = 3(10)^2 - 2(10) + 1 = 300 - 20 + 1 = 281\).
["# Understanding (a_{10} = 3(10)^2 - 2(10) + 1): A Step-by-Step Breakdown", "When tackling algebraic expressions, one key skill is evaluating expressions for specific variable values. A commonly seen format is (a_n = n^2 \ ext{(coefficient)} - n \ ext{(coefficient)} + \ ext{constant}). Let’s explore the calculation (a_{10} = 3(10)^2 - 2(10) + 1), showing how to simplify it step-by-step and arrive at the result: (a_{10} = 281).", "## What Does (a_{10} = 3(10)^2 - 2(10) + 1) Mean?", "This expression defines a quadratic formula substituted with (n = 10). It combines three parts:\n- A quadratic term: (3(10)^2), representing (3) times the square of (10)\n- A linear term: (-2(10)), which subtracts twice (10)\n- A constant: (+1), the final addition", "Let’s unfold the expression using order of operations (PEMDAS/BODMAS):", "---", "## Step 1: Calculate the Exponent and Multiplication", "Start with ( (10)^2 = 100 ).\nThen:\n[\n3(10)^2 = 3 \ imes 100 = 300\n]", "---", "## Step 2: Perform the Linear Term", "Next:\n[\n-2(10) = -20\n]", "---", "## Step 3: Combine All Terms", "Now plug back the calculated values:\n[\na_{10} = 300 - 20 + 1\n]", "Follow left to right:\n[\n300 - 20 = 280\n]\n[\n280 + 1 = 281\n]", "---", "## Final Result", "[\n\boxed{a_{10} = 281}\n]", "---", "## Why This Calculation Matters", "Expressions like (a_n = 3n^2 - 2n + 1) are fundamental in algebra, often used in sequences, polynomial modeling, and computer science algorithms. The ability to evaluate such expressions quickly enables deeper problem-solving in math and programming.", "Understanding each step — from exponentiation to multiplication and combination — ensures accuracy and builds confidence in tackling more complex formulas. Whether for homework, test prep, or coding, mastering algebraic evaluation is essential.", "---", "## Summary", "- Substitute (n = 10) into (a_n = 3n^2 - 2n + 1)\n- Compute (3(10)^2 = 300)\n- Compute (-2(10) = -20)\n- Add all parts: (300 - 20 + 1 = 281)\n- Final result: (\boxed{281})", "Keep practicing with different values of (n) to strengthen your algebra proficiency!"]
