第 \( n \) 項 \( a_n = S_n - S_{n-1} \) です。
["Title: Understanding the ( n )-th Term of a Sequence: ( a_n = S_n - S_{n-1} )", "In sequence analysis, one of the most fundamental concepts is the relationship between the ( n )-th term of a sequence and the partial sums. The expression ( a_n = S_n - S_{n-1} ) plays a crucial role in connecting partial sums to individual sequence terms. Whether you’re studying arithmetic sequences, geometric progressions, or more complex recursive sequences, understanding this formula is essential for deepening your grasp of mathematical patterns and summations.", "### What Are ( S_n ) and ( S_{n-1} )?", "First, define ( S_n ): it represents the ( n )-th partial sum of a sequence. Formally,", "[\nS_n = a_1 + a_2 + a_3 + \cdots + a_n\n]", "Similarly, ( S_{n-1} ) is the partial sum up to the ( (n-1) )-th term:", "[\nS_{n-1} = a_1 + a_2 + a_3 + \cdots + a_{n-1}\n]", "### Deriving the General Term ( a_n )", "To find ( a_n ), we compute the difference between these two partial sums:", "[\na_n = S_n - S_{n-1} = \left( \sum_{k=1}^{n} a_k \right) - \left( \sum_{k=1}^{n-1} a_k \right)\n]", "Subtracting the second sum from the first eliminates all terms from ( a_1 ) to ( a_{n-1} ), leaving only the ( n )-th term:", "[\na_n = S_n - S_{n-1} = a_n\n]", "This elegant identity allows us to recover the original sequence term directly from cumulative sum differences.", "### Applications Across Different Sequences", "This formula applies universally to both arithmetic and geometric sequences, as well as more intricate recursive sequences:", "- Arithmetic sequence with common difference ( d ):\n For ( a_n = d ), the partial sums follow ( S_n = \frac{n}{2}(2a_1 + (n-1)d) ), and the difference yields ( a_n = d ), confirming consistency.", "- Geometric sequence with ratio ( r ):\n Using ( a_n = ar^{n-1} ), the partial sums ( S_n = a \frac{1 - r^n}{1 - r} ) (for ( r <br/>\ne 1 )), and subtracting ( S_{n-1} ) recovers ( a_n = ar^{n-1} ).", "- General sequences defined by recurrence relations:\n Even for non-linear or complex recurrences, breaking sequence sums into differences helps verify or derive explicit formula terms.", "### Why This Formula Matters", "Understanding ( a_n = S_n - S_{n-1} ) empowers learners and mathematicians to:", "- Transition smoothly between cumulative (total) and instantaneous (current) values of a sequence.\n- Simplify proof techniques involving series convergence, telescoping sums, and recursive definitions.\n- Uncover hidden relationships in number theory, combinatorics, and calculus (e.g., in Taylor series or Riemann sums).", "### Conclusion", "The formula ( a_n = S_n - S_{n-1} ) is a cornerstone in the study of sequences and series. By mastering this relationship, you unlock the ability to determine each term from the cumulative sum pattern—providing clarity, precision, and deeper insight into the structure of mathematical sequences. Whether you’re solving problems or designing algorithms, this simple yet powerful identity remains indispensable.", "---", "Keywords: ( a_n = S_n - S_{n-1} ), partial sum, sequence term, summation, arithmetic sequence, geometric sequence, telescoping sum, mathematical sequences, series derivation, discrete mathematics, recurrence relations."]
