A geometric sequence has first term 4 and common ratio \( \frac{3}{2} \). What is the sum of the first 5 terms?

["A Geometric Sequence with First Term 4 and Common Ratio ( \frac{3}{2} ): Sum of the First 5 Terms", "Understanding geometric sequences is essential in mathematics, especially in topics like algebra, finance, and growth patterns. A geometric sequence is defined by a first term and a consistent multiplicative factor called the common ratio. In this article, we explore a specific geometric sequence where the first term is 4 and the common ratio is ( \frac{3}{2} ). We will also calculate the sum of the first five terms—a fundamental skill for students and learners alike.", "---", "### 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 number called the common ratio (( r )).\nFor example, if the first term ( a = 4 ) and ( r = \frac{3}{2} ), the sequence progresses like this:", "[\na_1 = 4\n]\n[\na_2 = 4 \cdot \frac{3}{2} = 6\n]\n[\na_3 = 6 \cdot \frac{3}{2} = 9\n]\n[\na_4 = 9 \cdot \frac{3}{2} = 13.5\n]\n[\na_5 = 13.5 \cdot \frac{3}{2} = 20.25\n]", "---", "### Formula for the Sum of the First ( n ) Terms", "The sum ( S_n ) of the first ( n ) terms of a geometric sequence can be calculated using the formula:", "[\nS_n = a \cdot \frac{r^n - 1}{r - 1} \quad \ ext{(for } r <br/>\ne 1\ ext{)}\n]", "Where:\n- ( a = 4 ) (first term)\n- ( r = \frac{3}{2} ) (common ratio)\n- ( n = 5 ) (number of terms)", "---", "### Calculating the Sum", "Plugging the values into the formula:", "[\nS_5 = 4 \cdot \frac{\left(\frac{3}{2}\right)^5 - 1}{\frac{3}{2} - 1}\n]", "First, calculate ( \left(\frac{3}{2}\right)^5 ):", "[\n\left(\frac{3}{2}\right)^5 = \frac{3^5}{2^5} = \frac{243}{32}\n]", "Next, compute the denominator:", "[\n\frac{3}{2} - 1 = \frac{1}{2}\n]", "Now substitute back:", "[\nS_5 = 4 \cdot \frac{\frac{243}{32} - 1}{\frac{1}{2}} = 4 \cdot \frac{\frac{243 - 32}{32}}{\frac{1}{2}} = 4 \cdot \frac{\frac{211}{32}}{\frac{1}{2}}\n]", "Dividing by ( \frac{1}{2} ) is the same as multiplying by 2:", "[\nS_5 = 4 \cdot \frac{211}{32} \cdot 2 = 4 \cdot \frac{211}{16} = \frac{844}{16} = 52.75\n]", "---", "### Final Result", "The sum of the first 5 terms of the geometric sequence with first term 4 and common ratio ( \frac{3}{2} ) is:", "[\n\boxed{52.75}\n]", "---", "### Why This Sum Matters", "Knowing how to compute sums in geometric sequences supports applications in compound interest, population modeling, and financial forecasting. With a common ratio greater than 1, this sequence grows quickly, and summing terms helps analyze total accumulation over discrete steps.", "Whether you’re student learning algebra, teacher preparing lessons, or someone exploring mathematical patterns, understanding geometric sequences like this one opens doors to deeper insights in quantitative reasoning.", "---", "### Summary", "- Geometric sequence: defined by a first term ( a ) and common ratio ( r ).\n- Common ratio ( r = \frac{3}{2} ), first term ( a = 4 ).\n- Use the sum formula: ( S_n = a \frac{r^n - 1}{r - 1} ).\n- Formula applied yields ( S_5 = 52.75 ).\n- This method is practical and widely applicable in both academic and real-world contexts.", "---", "Keywords: geometric sequence, sum of geometric sequence, common ratio ( \frac{3}{2} ), first term 4, sum of first 5 terms, geometric series calculation, algebra practice, educational math."]