So, \( S_6 = 3 \frac{2^6 - 1}{2 - 1} = 3 \times (64 - 1) = 3 \times 63 = 189 \).

["# Understanding the Geometric Series Formula: ( S_6 = 3 \frac{2^6 - 1}{2 - 1} = 189 )", "When tackling sequences and series, one powerful formula comes from the geometric series—especially useful when dealing with exponential growth patterns. A classic example illustrates this vividly:", "[ S_6 = 3 \frac{2^6 - 1}{2 - 1} = 3 \ imes (64 - 1) = 3 \ imes 63 = 189 ]", "## What Is This Formula About?", "The expression computes the sum of a geometric series where the first term is ( a = 3 ), the common ratio between terms is ( r = 2 ), and the sum covers the first 6 terms: ( S_6 ).", "### Breaking Down the Formula", "The geometric series sum formula is:\n[ S_n = a \frac{r^n - 1}{r - 1} \quad \ ext{for} \quad r <br/>\neq 1 ]", "- ( S_n ) = sum of the first ( n ) terms\n- ( a ) = first term\n- ( r ) = common ratio (each term multiplied by ( r ))\n- ( n ) = number of terms", "Plugging in:\n- ( a = 3 )\n- ( r = 2 )\n- ( n = 6 )", "[\nS_6 = 3 \ imes \frac{2^6 - 1}{2 - 1} = 3 \ imes (64 - 1) = 3 \ imes 63 = 189\n]", "### Why Is the Denominator ( 2 - 1 = 1 )?", "Because when the ratio ( r = 2 ), the denominator ( r - 1 ) becomes ( 1 ), simplifying the formula to:\n[\nS_6 = 3 \ imes (2^6 - 1) = 3 \ imes (64 - 1) = 189\n]\nThis shows how efficiently the formula streamlines calculations with repeated doubling.", "### Real-World Applications of This Sum", "This mathematical principle appears in:\n- Compound interest calculations where money grows exponentially each period\n- Computer science, especially in analyzing algorithms with recursive or branching exponential growth\n- Financial modeling, such as forecasting revenue with doubling sales trends", "### Tips for Computing Geometric Sums Easily", "- Verify ( r <br/>\neq 1 ) to avoid division errors.\n- Simplify exponents quickly—especially with powers like ( 2^6 = 64 ), which is foundational in binary-based systems.\n- Use the formula proactively to avoid tedious term-by-term addition.", "## Summary", "The formula ( S_6 = 3 \frac{2^6 - 1}{2 - 1} = 189 ) elegantly applies the geometric series formula to compute exponential growth patterns efficiently. Familiarizing yourself with such expressions empowers you to simplify complex calculations common in engineering, finance, and computer science.", "📌 Key Takeaway:\nThe formula ( S_n = a \frac{r^n - 1}{r - 1} ) is a powerful tool for summing geometric sequences where ( a = 3 ), ( r = 2 ), and ( n = 6 )—easily yielding ( S_6 = 189 ) in just a few steps.", "---", "Keywords: geometric series formula, sum of geometric sequence, (S_6 = 3 \frac{2^6 - 1}{2 - 1}), exponential growth calculation, mathematical formulas explained, computing series efficiently."]