\( a_n = S_n - S_{n-1} = (3n^2 + 5n) - (3n^2 - n - 2) = 6n + 2 \)
["# Understanding ( a_n = S_n - S_{n-1} = (3n^2 + 5n) - (3n^2 - n - 2) = 6n + 2 ): A Complete Breakdown", "Finding the general term ( a_n ) of a sequence defined as the difference between consecutive partial sums ( S_n ) is a powerful technique widely used in discrete mathematics and sequence analysis. In this article, we dive deep into the process of deriving ( a_n = S_n - S_{n-1} ), applying it specifically to the case where ( S_n = 3n^2 + 5n ), yielding ( a_n = 6n + 2 ). We explore step-by-step how this formula helps identify the original sequence and its real-world applications.", "---", "### What is ( a_n = S_n - S_{n-1} )?", "The expression ( a_n = S_n - S_{n-1} ) represents the difference between the ( n )-th and ( (n-1) )-th partial sums of a sequence. Since ( S_n = \sum_{k=1}^n a_k ), the term ( a_n ) corresponds precisely to the ( n )-th term of the sequence. This method is especially useful when partial sums ( S_n ) are given explicitly, enabling efficient extraction of individual terms.", "---", "### Step-by-Step Derivation: From ( S_n = 3n^2 + 5n ) to ( a_n = 6n + 2 )", "1. Given:\n Partial sum formula:\n [\n S_n = 3n^2 + 5n\n ]", "2. Compute ( S_{n-1} ):\n Replace ( n ) with ( n-1 ) in ( S_n ):\n [\n S_{n-1} = 3(n-1)^2 + 5(n-1)\n ]\n Expand:\n [\n S_{n-1} = 3(n^2 - 2n + 1) + 5n - 5 = 3n^2 - 6n + 3 + 5n - 5\n ]\n Simplify:\n [\n S_{n-1} = 3n^2 - n - 2\n ]", "3. Calculate ( a_n = S_n - S_{n-1} ):\n [\n a_n = (3n^2 + 5n) - (3n^2 - n - 2)\n ]\n Simplify the expression by combining like terms:\n [\n a_n = 3n^2 + 5n - 3n^2 + n + 2 = (3n^2 - 3n^2) + (5n + n) + 2 = 6n + 2\n ]", "---", "### The General Term of the Sequence", "We conclude that the general term is:\n[\na_n = 6n + 2\n]\nThis linear expression defines the ( n )-th term of the sequence derived from the quadratic partial sum. For example:\n- ( a_1 = 6(1) + 2 = 8 )\n- ( a_2 = 6(2) + 2 = 14 )\n- ( a_3 = 6(3) + 2 = 20 )\n... and so on.", "---", "### Why Is This Derivation Important?", "- Pattern Recognition: Understanding how difference equations translate partial sums into terms helps identify sequence patterns.\n- Simplifying Problems: Instead of summing recursively, computing terms directly from ( a_n ) speeds up calculations.\n- Applications: Such techniques are used in algorithm analysis, cumulative growth modeling, and signal processing where changes between states are critical.", "---", "### Real-World Applications", "- Economics: Modeling cumulative profit over time from incremental revenue and cost sums.\n- Physics: Calculating position or velocity changes from force or acceleration data.\n- Computer Science: Analyzing time complexity where ( a_n ) represents work done at step ( n ).", "---", "### Summary", "Deriving ( a_n ) via ( S_n - S_{n-1} ) is a foundational tool in sequence analysis. Starting from ( S_n = 3n^2 + 5n ), we systematically compute ( a_n ) to show it equals ( 6n + 2 ). This method not only simplifies sequence term identification but enhances problem-solving across disciplines involving cumulative data.", "Mastering this approach empowers students and professionals to decode complex sequences with confidence and precision.", "---", "### Key Takeaways\n- Use ( a_n = S_n - S_{n-1} ) to find the ( n )-th term from partial sums.\n- Expand and simplify algebraic expressions carefully to avoid errors.\n- Linear forms like ( 6n + 2 ) often emerge from quadratic sums.\n- This technique is essential in discrete math, applied sciences, and computational fields.", "---", "Elevate your sequence analysis skills — understanding ( a_n = S_n - S_{n-1} ) is your first step to unlocking elegant solutions in mathematics and beyond!"]
