A geometric sequence has a first term of 3 and a common ratio of 2. What is the sum of the first 6 terms?
["Understanding the Sum of a Geometric Sequence: First Term 3, Common Ratio 2", "When studying sequences, geometric sequences often stand out due to their exponential growth patterns. In this article, we’ll explore a specific geometric sequence with a clear first term and common ratio, and calculate the sum of its first six terms — a common yet essential calculation in mathematics, finance, and data science.", "### What is a Geometric Sequence?", "A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The general formula for the nth term of a geometric sequence is:", "[\na_n = a_1 \cdot r^{n-1}\n]", "where:\n- (a_1) is the first term,\n- (r) is the common ratio,\n- (n) is the term number.", "### Given Values", "For this problem:\n- First term ((a_1)) = 3\n- Common ratio ((r)) = 2\n- Number of terms ((n)) = 6", "### Step 1: List the First Six Terms", "Using the formula (a_n = 3 \cdot 2^{n-1}), calculate each term:\n- (a_1 = 3 \cdot 2^{0} = 3 \cdot 1 = 3)\n- (a_2 = 3 \cdot 2^{1} = 3 \cdot 2 = 6)\n- (a_3 = 3 \cdot 2^{2} = 3 \cdot 4 = 12)\n- (a_4 = 3 \cdot 2^{3} = 3 \cdot 8 = 24)\n- (a_5 = 3 \cdot 2^{4} = 3 \cdot 16 = 48)\n- (a_6 = 3 \cdot 2^{5} = 3 \cdot 32 = 96)", "The sequence is: 3, 6, 12, 24, 48, 96", "### Step 2: Use the Geometric Series Sum Formula", "To efficiently compute the sum of the first (n) terms, use the geometric series sum formula:", "[\nS_n = a_1 \cdot \frac{r^n - 1}{r - 1}\n]", "Plug in the known values:", "[\nS_6 = 3 \cdot \frac{2^6 - 1}{2 - 1} = 3 \cdot \frac{64 - 1}{1} = 3 \cdot 63 = 189\n]", "### Final Answer", "The sum of the first six terms of the geometric sequence with first term 3 and common ratio 2 is 189.", "### Why This Matters", "Understanding how to sum geometric sequences helps solve problems in compound interest, population growth models, and algorithmic time complexity analysis. Mastering the sum formula empowers students and professionals alike to work efficiently with exponential patterns.", "---", "Keywords: geometric sequence sum, geometric series formula, first term 3, common ratio 2, sum of first 6 terms, exponential sum formula, finite geometric progression, math problem solving."]
