\( S_5 = 4 \cdot \frac{(\frac{3}{2})^5 - 1}{\frac{3}{2} - 1} = 4 \cdot \frac{\frac{243}{32} - 1}{\frac{1}{2}} = 4 \cdot \frac{\frac{211}{32}}{\frac{1}{2}} = 4 \cdot \frac{211}{32} \cdot 2 = 8 \cdot \frac{211}{32} = \frac{1688}{32} = 52.75 \).

["Understanding the Combinatorial Power: Decoding ( S_5 = 4 \cdot \frac{(\frac{3}{2})^5 - 1}{\frac{3}{2} - 1} ) Step-by-Step", "The expression ( S_5 = 4 \cdot \frac{(\frac{3}{2})^5 - 1}{\frac{3}{2} - 1} ) may look complex at first glance, but with a clear breakdown, it reveals elegant mathematical reasoning rooted in geometric series principles. This equation not only highlights algebraic computation but also connects deeply with foundational concepts in combinatorics and series summation. In this article, we’ll explore how this expression evaluates to 52.75 step-by-step, shedding light on its significance.", "### The Core Formula: A Geometric Series in Disguise", "At its core, the expression ( S_5 ) leverages the formula for the sum of a geometric series, a fundamental tool in mathematics with applications across probability, finance, and beyond.", "Start from the general formula for a finite geometric series:", "[\n\sum_{k=0}^{n-1} r^k = \frac{r^n - 1}{r - 1}, \quad \ ext{for } r <br/>\ne 1\n]", "Here, ( r = \frac{3}{2} ) and the sum spans integer powers from ( k = 0 ) to ( k = 4 ), which matches the 5 terms in ( S_5 ):", "[\n\frac{(\frac{3}{2})^5 - 1}{\frac{3}{2} - 1}\n]", "But in the equation, this sum is multiplied by 4:", "[\nS_5 = 4 \cdot \frac{(\frac{3}{2})^5 - 1}{\frac{3}{2} - 1}\n]", "### Step 1: Compute ( (\frac{3}{2})^5 )", "We begin computing ( \left(\frac{3}{2}\right)^5 ):", "[\n\left(\frac{3}{2}\right)^5 = \frac{3^5}{2^5} = \frac{243}{32}\n]", "This matches the intermediate result shown: ( \frac{243}{32} ), which comes from expanding ( \left(\frac{3}{2}\right)^5 ).", "### Step 2: Subtract 1 from the Power", "Next, subtract 1 (equivalent to ( (\frac{3}{2})^0 )):", "[\n\frac{243}{32} - 1 = \frac{243}{32} - \frac{32}{32} = \frac{211}{32}\n]", "This matches the next step in the factorization: ( \frac{\frac{211}{32}}{\frac{1}{2}} ).", "### Step 3: Divide by ( \frac{3}{2} - 1 = \frac{1}{2} )", "Now divide the result by ( \frac{1}{2} ):", "[\n\frac{\frac{211}{32}}{\frac{1}{2}} = \frac{211}{32} \cdot 2 = \frac{422}{32}\n]", "This simplifies to ( \frac{211}{16} ), but note—keeping it as a fraction for clarity before final multiplication improves precision.", "### Step 4: Multiply by 4", "Now multiply the entire expression by 4:", "[\nS_5 = 4 \cdot \frac{211}{32} = \frac{844}{32}\n]", "Wait—evaluating ( 4 \cdot \frac{211}{32} = \frac{844}{32} = 26.375)? No! There’s a hidden simplification.", "Let’s retrace:", "From earlier:", "[\nS_5 = 4 \cdot \frac{\frac{211}{32}}{\frac{1}{2}} = 4 \cdot \frac{211}{32} \cdot 2 = 8 \cdot \frac{211}{32}\n]", "This step skips intermediate division by separate fractions and directly applies:", "[\n4 \cdot \left( \left(\frac{3}{2}\right)^5 - 1 \right) \cdot \left( \frac{2}{3/2 - 1} \right) = 4 \cdot \left( \frac{243}{32} - 1 \right) \cdot 2\n]", "So:", "[\n= 4 \cdot 2 \cdot \left( \frac{243 - 32}{32} \right) = 8 \cdot \frac{211}{32} = \frac{1688}{32}\n]", "### Final Evaluation: ( \frac{1688}{32} = 52.75 )", "Now divide: ( 1688 \div 32 )", "Break it down:", "[\n32 \ imes 52 = 1664,\quad 1688 - 1664 = 24,\quad \frac{24}{32} = 0.75\n]", "Thus:", "[\n\frac{1688}{32} = 52 + 0.75 = 52.75\n]", "---", "### Why This Matters: From Formula to Application", "The expression ( S_5 = 4 \cdot \frac{(\frac{3}{2})^5 - 1}{\frac{3}{2} - 1} ) mirrors the sum ( 1 + r + r^2 + r^3 + r^4 ) with ( r = \frac{3}{2} ), scaled by 4. While not a standard combinatorial count, its structure echoes formulas used in weighted averages, repeated dynamical systems, or geometric growth models—contexts where powers and ratios naturally emerge.", "Furthermore, the simplicity of fractional arithmetic underscores the elegance of expressing real-world multiplicative growth or risk calculations succinctly. In fields like finance (e.g., compound growth over 5 periods at 50% growth rate) or data science, similar expressions factored neatly into algorithms and predictions.", "---", "### Summary", "- ( S_5 ) stems from a geometric series sum scaled by 4\n- Computation follows: ( \left(\frac{3}{2}\right)^5 = \frac{243}{32} ), minus 1 gives ( \frac{211}{32} )\n- Dividing by ( \frac{1}{2} ) is equivalent to multiplying by 2: ( \frac{422}{32} )\n- Final multiplication by 4 yields ( \frac{1688}{32} = 52.75 )", "This breakdown not only confirms the numeric value but reveals a powerful tool for evaluating complex series efficiently—proving that even abstract formulas carry precise, real-world interpretable outcomes.", "---", "Key Takeaway: Mastering expressions like ( S_5 ) deepens fluency in applying geometric series, simplifying algebraic reasoning—a skill vital across mathematics, science, and engineering applications.", "---", "Keywords: ( S_5 = 4 \cdot \frac{(\frac{3}{2})^5 - 1}{\frac{3}{2} - 1} ), geometric series, fractional arithmetic, ( (\frac{3}{2})^5 ), computation breakdown, algebra simplification, 52.75 value, combinatorial power, step-by-step math, series summation."]