\( S_{n-1} = 3(n-1)^2 + 5(n-1) = 3(n^2 - 2n + 1) + 5n - 5 = 3n^2 - 6n + 3 + 5n - 5 \) です。

["# Understanding ( S_{n-1} = 3(n-1)^2 + 5(n-1) ): A Complete Breakdown", "Mathematics often involves recursive or shifted sequences, where understanding indexing shifts like ( S_{n-1} ) is essential for problem-solving and pattern recognition. One such algebraic expression frequently encountered is:", "[\nS_{n-1} = 3(n-1)^2 + 5(n-1)\n]", "In this article, we’ll expand, simplify, and analyze this expression step-by-step, helping you master operations on quadratic sequences and improve your algebraic fluency.", "## Why Understanding ( S_{n-1} ) Matters", "Shifts in index notation—like replacing ( n ) with ( n-1 )—are common in recurrence relations, summation formulas, and sequence analysis. Recognizing how expressions change under index shifts enables deeper insight into iterative processes, summation techniques, and polynomial behavior.", "So, let’s unpack ( S_{n-1} ) fully.", "---", "## Step 1: Expand the Expression", "Start by expanding ( (n-1)^2 ) and distributing:", "[\nS_{n-1} = 3(n-1)^2 + 5(n-1)\n]", "First, expand ( (n-1)^2 ):", "[\n(n-1)^2 = n^2 - 2n + 1\n]", "Now multiply by 3:", "[\n3(n^2 - 2n + 1) = 3n^2 - 6n + 3\n]", "Next, expand ( 5(n-1) ):", "[\n5(n-1) = 5n - 5\n]", "Now combine both parts:", "[\nS_{n-1} = 3n^2 - 6n + 3 + 5n - 5\n]", "---", "## Step 2: Combine Like Terms", "Group the like terms—those with ( n^2 ), ( n ), and constant components:", "- Quadratic term: ( 3n^2 )\n- Linear terms: ( -6n + 5n = -n )\n- Constant terms: ( 3 - 5 = -2 )", "Putting it all together:", "[\nS_{n-1} = 3n^2 - n - 2\n]", "---", "## Step 3: Interpret the Result", "We transformed the original shifted expression into a standard quadratic form:", "[\nS_{n-1} = 3n^2 - n - 2\n]", "This reveals that ( S_{n-1} ) is a parabola opening upwards (since the coefficient of ( n^2 ) is positive), shifted left by one index. Understanding this directly impacts how you compute terms or sum sequences starting at ( n-1 ).", "---", "## Why This Transformation Is Useful", "- Simplification: You can now work with a cleaner, expanded quadratic rather than nested parentheses.\n- Summation & Aggregation: When computing sums like ( \sum_{k=1}^{n-1} S_k ), this simplified form is easier to apply.\n- Recurrence Relations: Identifying such shifts helps in solving recurrence equations by recognizing patterns and base cases.\n- Graphical Insight: Knowing ( S_{n-1} ) in standard form reveals its vertex, minimum point, and growth behavior.", "---", "## Practical Example: Computing ( \sum_{k=1}^{n-1} S_k )", "Suppose you want to compute the sum:", "[\n\sum_{k=1}^{n-1} S_k\n]", "Using ( S_k = 3k^2 - k - 2 ), we write:", "[\n\sum_{k=1}^{n-1} S_k = \sum_{k=1}^{n-1} (3k^2 - k - 2) = 3\sum_{k=1}^{n-1} k^2 - \sum_{k=1}^{n-1} k - 2\sum_{k=1}^{n-1} 1\n]", "Now apply standard summation formulas:", "- ( \sum_{k=1}^{m} k^2 = \frac{m(m+1)(2m+1)}{6} )\n- ( \sum_{k=1}^{m} k = \frac{m(m+1)}{2} )\n- ( \sum_{k=1}^{m} 1 = m )", "With ( m = n-1 ):", "[\n\sum_{k=1}^{n-1} S_k = 3 \cdot \frac{(n-1)n(2n-1)}{6} - \frac{(n-1)n}{2} - 2(n-1)\n]", "Simplify each term:", "1. Quadratic sum:\n[\n3 \cdot \frac{(n-1)n(2n-1)}{6} = \frac{(n-1)n(2n-1)}{2} = \frac{2n^3 - 3n^2 + n}{2}\n]", "2. Linear sum:\n[\n\frac{(n-1)n}{2} = \frac{n^2 - n}{2}\n]", "3. Constant term:\n[\n2(n-1) = 2n - 2\n]", "Now combine:", "[\n\sum_{k=1}^{n-1} S_k = \frac{2n^3 - 3n^2 + n}{2} - \frac{n^2 - n}{2} - (2n - 2)\n]", "Combine fractions:", "[\n= \frac{(2n^3 - 3n^2 + n) - (n^2 - n)}{2} - 2n + 2 = \frac{2n^3 - 4n^2 + 2n}{2} - 2n + 2\n]", "Simplify:", "[\n= n^3 - 2n^2 + n - 2n + 2 = n^3 - 2n^2 - n + 2\n]", "Thus, verified:", "[\n\sum_{k=1}^{n-1} S_k = n^3 - 2n^2 - n + 2\n]", "---", "## Conclusion", "The expression ( S_{n-1} = 3(n-1)^2 + 5(n-1) ) simplifies beautifully to:", "[\nS_{n-1} = 3n^2 - n - 2\n]", "This form unlocks easier computation in summation, recurrence, and analysis. Mastering such index shifts and expansions is invaluable in algebra, discrete math, and algorithm analysis.", "If you encounter expressions involving ( n-1 ), remember to expand, combine like terms, and restructure—turning nested forms into powerful polynomial tools.", "---", "### Key Takeaways", "- Expand ( (n-1)^2 ) and distribute carefully.\n- Combine like terms directly.\n- Recognize simplified forms reveal deeper structure.\n- Use expansions to compute sums, recurrences, and analyze function behavior.", "Keep practicing shifting indices—your algebraic agility will greatly improve!"]