Solution: The given sequence is an arithmetic sequence with the first term \( a = 3 \) and common difference \( d = 5 \). The general term of the sequence is given by:
["Understanding the General Term of an Arithmetic Sequence: A Step-by-Step Guide", "Arithmetic sequences are fundamental in mathematics and play a crucial role in various applications, from simple pattern recognition to complex problem-solving in algebra and beyond. If you’re encountering a sequence defined by a first term and a common difference, knowing how to find its general term is essential. In this article, we’ll explore the solution for an arithmetic sequence with first term ( a = 3 ) and common difference ( d = 5 ), and explain how to derive the general term formula.", "---", "### What Is an Arithmetic Sequence?", "An arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a constant value, known as the common difference ( d ). For example, if the first term is ( a ), then the sequence continues:", "[\na, a + d, a + 2d, a + 3d, \ldots\n]", "---", "### Given Sequence: Explicit Parameters", "In the specific case we’re analyzing:\n- First term: ( a = 3 )\n- Common difference: ( d = 5 )", "Thus, the sequence starts like:\n3, 8, 13, 18, 23, …", "---", "### Deriving the General Term of the Sequence", "The general term ( T_n ) of an arithmetic sequence can be expressed using the position ( n ) of the term in the sequence. The formula is:", "[\nT_n = a + (n - 1)d\n]", "Where:\n- ( T_n ) = the ( n^{\ ext{th}} ) term\n- ( a ) = first term\n- ( d ) = common difference\n- ( n ) = term position (must be a positive integer)", "Substituting the given values:", "[\nT_n = 3 + (n - 1) \ imes 5\n]", "---", "### Step-by-Step Simplification", "Let’s simplify the expression to make it easier to use:", "[\nT_n = 3 + 5(n - 1)\n]", "Distribute the 5:", "[\nT_n = 3 + 5n - 5\n]", "Combine like terms:", "[\nT_n = 5n - 2\n]", "---", "### Final Answer: General Term Formula", "The general term of the given arithmetic sequence is:", "[\n\boxed{T_n = 5n - 2}\n]", "---", "### How to Use the General Term", "Using ( T_n = 5n - 2 ), you can directly compute any term in the sequence by plugging in the desired value of ( n ). For example:", "- To find the 4th term:\n ( T_4 = 5(4) - 2 = 20 - 2 = 18 )", "This matches the term we listed earlier.", "---", "### Why the General Term Matters", "- Predictability: Find any term without listing all prior terms\n- Efficiency: Solve for ( n ) when the term is given\n- Applications: Useful in modeling linear growth, algebra problems, and real-world data patterns", "---", "### Summary", "For the arithmetic sequence with first term ( a = 3 ) and common difference ( d = 5 ), the general term is succinctly given by:", "[\nT_n = 5n - 2\n]", "Understanding and applying this formula opens up powerful tools for analyzing and working with sequences in mathematics and beyond. Whether solving equations, plotting graphs, or exploring patterns, mastering the general term prepares you for more advanced concepts with confidence.", "---", "Keywords: arithmetic sequence general term, arithmetic sequence formula, find nth term, arithmetic sequence first term, arithmetic sequence common difference, formula Tₙ = 5n − 2"]
