Thus, the sequence has first term \( a_1 = 3^1 - 1 = 2 \) and common ratio \( r = 3 \).

["Title: The Closed Form of a Geometric Sequence With First Term 2 and Ratio 3", "Meta Description:\nDiscover the explicit formula for the geometric sequence starting at ( a_1 = 2 ) with a common ratio of 3. Learn how to derive and use ( a_n = 3^{n-1} ) in math, programming, and real-world applications.", "---", "### Introduction", "In mathematics, geometric sequences are fundamental building blocks for modeling growth, decay, and recurring patterns. A defining feature of such sequences is their common ratio—the factor by which each term multiplies to get the next. In this article, we explore a specific geometric sequence with a clean, exponential structure:\nFirst term: ( a_1 = 2 )\nCommon ratio: ( r = 3 )", "We’ll break down how to derive its explicit formula, explain the role of the first term and ratio, and highlight practical uses in science, finance, and computer science.", "---", "### Understanding the Geometric Sequence", "A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. For a sequence defined as:\n[\na_n = a_1 \cdot r^{n-1}\n]\n- ( a_n ) is the ( n )th term\n- ( a_1 ) is the first term\n- ( r ) is the common ratio\n- ( n ) is the term position (a positive integer)", "### Given Sequence Parameters", "For the sequence described:\n- The first term is ( a_1 = 3^1 - 1 = 2 )\n- The common ratio is ( r = 3 )", "Thus, the explicit formula for this sequence is:\n[\na_n = 2 \cdot 3^{n-1}\n]", "---", "### Deriving the Formula", "Let’s verify and explain how this formula arises.", "The general term of a geometric sequence is defined recursively:\n[\na_1 = 2\n]\n[\na_n = a_{n-1} \cdot 3 \quad \ ext{for } n \geq 2\n]", "Unfolding the recursion:\n[\na_2 = a_1 \cdot 3 = 2 \cdot 3\n]\n[\na_3 = a_2 \cdot 3 = (2 \cdot 3) \cdot 3 = 2 \cdot 3^2\n]\n[\na_4 = a_3 \cdot 3 = (2 \cdot 3^2) \cdot 3 = 2 \cdot 3^3\n]", "Observing the pattern:\n- ( a_1 = 2 \cdot 3^0 )\n- ( a_2 = 2 \cdot 3^1 )\n- ( a_3 = 2 \cdot 3^2 )\n- ( \vdots )\n- ( a_n = 2 \cdot 3^{n-1} )", "This confirms the explicit formula:\n[\n\boxed{a_n = 2 \cdot 3^{n-1}}\n]", "---", "### Key Features of This Sequence", "- Exponential Growth: Since ( r = 3 > 1 ), the sequence grows rapidly.\n- Starts At 2: Unlike sequences starting at 1, ( a_n ) never reaches zero.\n- Cleaner Closed Form: The formula combines the initial value and ratio efficiently.", "---", "### Applications of Geometric Sequences", "#### 1. Mathematical Modeling\nGeometric sequences model phenomena with consistent multiplicative growth or decay, such as:\n- Population growth, bacterial division (doubling every hour)\n- Radioactive decay (halving over times multiples)\n- Compound interest calculations", "#### 2. Computer Science\nUsed in algorithm analysis to calculate runtime in recursive processes with exponential branching, like certain divide-and-conquer algorithms or binary tree traversals.", "#### 3. Finance and Economics\nIdeal for modeling investments with compound growth:\n[\nA = P(1 + r)^n\n]\nWhere ( P ) is principal, ( r ) rate, and ( n ) time periods.", "---", "### Programmatic Implementation", "To compute terms efficiently in code (e.g., Python):", "python\ndef geometric_sequence(n):\n a1 = 2\n r = 3\n return a1 * (r ** (n - 1))", "Example:\npython\nprint(geometric_sequence(1)) # Output: 2\nprint(geometric_sequence(4)) # Output: 54 (2 × 3³)", "This formula minimizes computation time compared to iterative multiplication, especially for large ( n ).", "---", "### Final Thoughts", "The geometric sequence with ( a_1 = 2 ) and ( r = 3 ) is a prime example of exponential behavior governed by simple rules. Its explicit formula, ( a_n = 2 \cdot 3^{n-1} ), encapsulates infinite patterns in finite notation—powerful for math, science, and technology. Whether analyzing natural growth or optimizing algorithms, mastering such sequences unlocks deeper analytical insight.", "---", "Sync Keywords: geometric sequence formula, explicit formula derivation, ( a_n = 3^{n-1} ), exponential growth, math patterns, recursive sequences, compound interest, algorithm analysis, Python geometric sequence.", "---", "Read More:\n- Complete guide to geometric sequences\n- How to solve arithmetic vs geometric sequences\n- Applications of exponential functions in real life", "---", "Internal Link Suggestions:\n- Link to “Geometric vs. Arithmetic Sequences”\n- Link to “Exponential Functions and Real-World Growth”\n- Link to “Implementing Geometric Sequences in Python”", "---", "Optimization Tips:\n- Use schema markup for formula lists (e.g., Schema.org Math vocabulary)\n- Create a focused FAQ on “How to find closed-form of geometric sequences”\n- Include tables comparing growth rates of different ( r ) values", "---", "Keywords (SEO Target):\n( a_1 = 2 ), ( r = 3 ), geometric sequence explicit formula, geometric term formula, ( a_n = 2 \cdot 3^{n-1} )", "---", "By understanding and applying this foundational sequence, anyone can deepen their grasp of exponential patterns—essential for advanced mathematics and modern problem-solving."]