a_n = 2 + (n - 1) \times 3 = 3n - 1

["Understanding the Linear Formula: aₙ = 2 + (n − 1) × 3 = 3n − 1", "When exploring sequences in mathematics, identifying patterns and expressing them as formulas is essential. One common formula that arises is the arithmetic sequence defined as:", "aₙ = 2 + (n − 1) × 3", "This equation describes a linear sequence where each term increases consistently. But how can we simplify and understand it better? Let’s break it down.", "### Breaking Down the Formula", "The given formula is:", "aₙ = 2 + (n − 1) × 3", "First, interpret what each part represents:", "- n: The position of the term in the sequence (n = 1, 2, 3, ...).\n- (n − 1): The term’s offset from the first term (since sequences start at n = 1).\n- × 3: The common difference between consecutive terms.\n- + 2: The initial value when n = 1.", "### Simplifying the Expression", "Let’s simplify the right-hand side to recognize its structure:", "[\naₙ = 2 + (n - 1) \ imes 3\n= 2 + 3n - 3\n= 3n - 1\n]", "So, the explicit formula for the n-th term is:", "aₙ = 3n − 1", "This form makes it easier to compute any term in the sequence by simply substituting the value of n.", "### What Is This Sequence?", "Substituting small values of n shows the pattern:", "- When n = 1:\n a₁ = 3(1) − 1 = 2\n- When n = 2:\n a₂ = 3(2) − 1 = 5\n- When n = 3:\n a₃ = 3(3) − 1 = 8\n- When n = 4:\n a₄ = 3(4) − 1 = 11", "So the sequence begins: 2, 5, 8, 11, …", "This is an arithmetic sequence with a common difference (d) of 3, meaning each term increases by 3 from the previous one.", "### Why Is the Linear Formula Useful?", "Arithmetic sequences modeled by linear formulas like aₙ = 3n − 1 allow for:", "- Efficient term calculation without having to list all previous terms.\n- Predictive modeling—quickly determining any term’s value.\n- Algorithm and programming applications, such as generating progressions for loops or financial models (e.g., linearly growing income or savings).", "### Applications in Real Life", "Linear formulas like aₙ = 3n − 1 are widely used across disciplines:", "- Education finance: Modeling linear growth of savings or graduated payments.\n- Physics: Calculating position over time in motion with constant velocity.\n- Computer science: Iterative algorithms where values increase uniformly.", "### Final Thoughts", "The formula aₙ = 2 + (n − 1) × 3 simplifies elegantly to aₙ = 3n − 1, revealing a clean arithmetic sequence with a consistent step size. Understanding and using such formulas empowers problem-solving in mathematics and related fields, enabling quick predictions and insightful pattern recognition.", "---", "Key Takeaways:", "- aₙ = 2 + (n − 1) × 3 is the recursive formula for the sequence.\n- Simplifying gives aₙ = 3n − 1, an explicit formula.\n- The sequence increases by 3 with each term; it’s arithmetic.\n- Linear formulas support efficient computation and modeling.\n- Practical uses span finance, science, programming, and education.", "---", "Master this simple yet powerful formula to unlock deeper insights into sequences and their applications!"]