\( a_1 = 3 \), \( a_3 = 3r^2 = 27 \Rightarrow r^2 = 9 \Rightarrow r = 3 \) (assuming positive)
["Understanding the Sequence: Deriving the Common Ratio ( r ) Given ( a_1 = 3 ) and ( a_3 = 27 )", "In the study of geometric sequences, identifying the common ratio is fundamental to understanding the pattern and predicting future terms. Given basic facts about a sequence—specifically, the first term ( a_1 = 3 ) and the third term ( a_3 = 27 )—we can systematically deduce the common ratio ( r ), assuming it is positive.", "### Step 1: Recall the General Formula\nFor a geometric sequence, each term is generated by multiplying the previous term by a constant called the common ratio ( r ). The ( n )-th term is expressed as:\n[\na_n = a_1 \cdot r^{n-1}\n]\nThis formula applies for any positive integer ( n ).", "### Step 2: Apply the Formula to the Third Term\nWe are given ( a_3 = 27 ) and ( a_1 = 3 ). Using the formula:\n[\na_3 = a_1 \cdot r^{3-1} = a_1 \cdot r^2\n]\nSubstitute the known values:\n[\n27 = 3 \cdot r^2\n]", "### Step 3: Solve for ( r^2 )\nDivide both sides by 3 to isolate ( r^2 ):\n[\nr^2 = \frac{27}{3} = 9\n]", "### Step 4: Solve for ( r ) (Assuming Positive Ratio)\nSince the problem specifies to assume ( r > 0 ), we take the positive square root:\n[\nr = \sqrt{9} = 3\n]", "### Conclusion\nBy applying the geometric sequence formula step-by-step, we determine that the common ratio ( r ) in this sequence is ( 3 ). This positive value ensures consistency and predictability in further calculations, enabling accurate determination of any term in the sequence. For example, the second term is:\n[\na_2 = a_1 \cdot r = 3 \cdot 3 = 9\n]\nContinuing, the full sequence begins: ( 3,\ 9,\ 27,\ 81,\ 243,) showing clear geometric growth with ratio ( 3 ).", "Understanding the common ratio empowers students and learners to analyze and extend geometric sequences confidently, forming a foundational skill in algebra and mathematical reasoning.", "---", "Keywords: geometric sequence, common ratio, ( a_1 = 3 ), ( a_3 = 27 ), algebraic derivation, positive ratio, mathematical sequence, solving for ( r )"]
