Solution: The sequence is arithmetic with first term $a_1 = 2$ and common difference $d = 3$. The $n$-th term is:
["The Arithmetic Sequence Formula: Finding the $n$-th Term When $a_1 = 2$ and $d = 3$", "Understanding arithmetic sequences is fundamental to mastering algebra and discovering patterns in mathematics. An arithmetic sequence is defined by a constant difference, known as the common difference $d$, between consecutive terms. In this article, we explore the standard solution form for the $n$-th term of an arithmetic sequence and specifically solve the case where the first term is $a_1 = 2$ and the common difference $d = 3$.", "---", "### What Is an Arithmetic Sequence?", "An arithmetic sequence is a list of numbers where each term increases by a fixed value called the common difference $d$. For example, starting with $a_1 = 2$ and $d = 3$, the sequence unfolds as:", "$$\n2,\ 5,\ 8,\ 11,\ 14,\ \dots\n$$", "This predictable increase makes it easy to calculate any term without calculating all previous ones.", "---", "### The General Formula for the $n$-th Term", "The $n$-th term $a_n$ of an arithmetic sequence is given by:", "$$\na_n = a_1 + (n - 1)d\n$$", "Where:\n- $a_1$ is the first term\n- $d$ is the common difference\n- $n$ is the term number (a positive integer)", "This formula works because each term adds the common difference $d$ a total of $n - 1$ times to get to the $n$-th term:", "$$\na_n = a_1 + d + d + \dots + d = a_1 + (n - 1)d\n$$", "---", "### Applying the Formula to $a_1 = 2$ and $d = 3$", "Substitute the known values into the general formula:", "$$\na_n = 2 + (n - 1) \cdot 3\n$$", "Simplify the expression:", "$$\na_n = 2 + 3n - 3 = 3n - 1\n$$", "---", "### Final Answer", "The $n$-th term of the arithmetic sequence with $a_1 = 2$ and $d = 3$ is:", "$$\n\boxed{a_n = 3n - 1}\n$$", "---", "### Why This Formula Matters", "Knowing how to derive and apply the $n$-th term formula empowers students and learners to quickly analyze linear patterns, solve real-world problems involving constant growth, and build deeper algebraic intuition. Whether in finance, physics, or computer science, arithmetic sequences provide a clear, structured approach to modeling stepwise change.", "Remember: identifying $a_1$ and $d$ is the first step — then use the simple formula to unlock any term instantly."]
