Solution:** The sequence 3, 9, 15, 21, ... is arithmetic with first term \( a = 3 \) and common difference \( d = 6 \).

["Solution: Understanding the Arithmetic Sequence 3, 9, 15, 21, ... with First Term ( a = 3 ) and Common Difference ( d = 6 )", "Arithmetic sequences are fundamental patterns in mathematics that appear across various real-world applications—from finance and science to computer algorithms and daily planning. One of the most well-known arithmetic sequences begins with the terms 3, 9, 15, 21, ..., demonstrating a clear and predictable structure.", "### What Defines an Arithmetic Sequence?", "An arithmetic sequence is a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is called the common difference, denoted by ( d ). The sequence is defined formally by its first term ( a ) and common difference ( d ).", "### The Given Sequence Explained", "The sequence 3, 9, 15, 21, ... is an arithmetic sequence with:\n- First term ( a = 3 )\n- Common difference ( d = 6 )", "Each term increases by 6 from the previous one:\n- ( 3 + 6 = 9 )\n- ( 9 + 6 = 15 )\n- ( 15 + 6 = 21 )\n- ( 21 + 6 = 27 ), and so on.", "### General Formula for the nth Term", "To find any term in the sequence, use the formula for the ( n )th term of an arithmetic sequence:\n[\na_n = a + (n - 1)d\n]", "For this sequence:\n[\na_n = 3 + (n - 1) \ imes 6\n]\n[\na_n = 3 + 6n - 6 = 6n - 3\n]", "This formula allows you to compute the value at any position ( n ) in the sequence efficiently.", "### Why This Sequence Matters", "Understanding arithmetic sequences like 3, 9, 15, 21, ... builds a foundation in pattern recognition and mathematical reasoning. Real-world applications include:\n- Saving plans where fixed amounts are added monthly\n- Scheduling tasks that occur at regular intervals\n- Data analysis where trends show consistent change", "### Summing the Sequence", "If you want to calculate the sum of the first ( n ) terms, the sum ( S_n ) is given by:\n[\nS_n = \frac{n}{2} (a + a_n)\n]\nSubstituting ( a_n = 6n - 3 ):\n[\nS_n = \frac{n}{2} (3 + (6n - 3)) = \frac{n}{2} (6n) = 3n^2\n]", "This elegant result shows that the sum of the first ( n ) terms grows quadratically, highlighting the power of arithmetic series.", "### Conclusion", "The sequence 3, 9, 15, 21, ... is a classic example of an arithmetic progression with a clear common difference of 6. Mastering such sequences strengthens analytical skills and supports deeper exploration in algebra and beyond. Whether for homework, studies, or practical applications, recognizing and applying arithmetic patterns empowers effective problem-solving.", "---", "Keywords: arithmetic sequence, 3, 9, 15, 21, common difference, first term, general formula, arithmetic progression, sum of terms, ( a = 3 ), ( d = 6 ), algebra, pattern recognition"]