Substitute \( n = 5 \) into the formula: \( a_5 = 3(5)^2 - 2(5) + 1 \).
["Optimizing the Formula: Calculating ( a_5 ) with Substitute ( n = 5 )", "When working with quadratic sequences or polynomial expressions in discrete mathematics, substitution of specific values like ( n = 5 ) is essential to evaluate the expression accurately. One such formula frequently studied in algebra is:", "[\na_n = 3n^2 - 2n + 1\n]", "This formula appears commonly in sequences where each term is determined by a quadratic expression in ( n ). Substituting ( n = 5 ) allows us to compute the fifth term, ( a_5 ), which plays a critical role in analyzing sequence behavior and testing algebraic manipulation skills.", "---", "### Step-by-Step Calculation of ( a_5 )", "Let’s substitute ( n = 5 ) directly into the formula:", "[\na_5 = 3(5)^2 - 2(5) + 1\n]", "Start by evaluating the power:", "[\n5^2 = 25\n]", "Then multiply by the coefficient:", "[\n3 \ imes 25 = 75\n]", "Next, compute the linear term:", "[\n-2 \ imes 5 = -10\n]", "Now combine all parts:", "[\na_5 = 75 - 10 + 1\n]", "Simplify the arithmetic:", "[\n75 - 10 = 65\n]", "[\n65 + 1 = 66\n]", "---", "### ✅ Final Result", "[\n\boxed{a_5 = 66}\n]", "---", "### Why This Substitution Matters", "Substituting known values into formulas like ( a_n = 3n^2 - 2n + 1 ) is more than just arithmetic—it helps verify correctness when deriving behavior patterns, validating recursive relationships, or solving real-world problems involving discrete data. Moreover, understanding such evaluations builds a strong foundation in algebraic thinking and sequence analysis.", "In educational contexts—from high school math to early college courses—this practice reinforces accurate computation, promotes algebraic fluency, and supports deeper comprehension of how variables affect outcomes in formulas.", "---", "### Summary", "- Substituting ( n = 5 ) into ( a_n = 3n^2 - 2n + 1 ) gives ( a_5 = 66 ).\n- The calculation follows standard order of operations: square first, then multiplication, followed by addition and subtraction.\n- Mastering such substitutions empowers precise evaluation critical in intermediate algebra and beyond.", "If you're studying sequences, series, or quadratic expressions, regularly practicing substitutions like ( n = 5 ) strengthens your mathematical toolkit and helps you confidently decode more complex formulas."]
