The first term \( a = 3 \) and the common ratio \( r = rac{1.5}{3} = 0.5 \).

["# Exploring the First Term and Common Ratio: An Introduction to Geometric Sequences", "Understanding geometric sequences is essential in mathematics, finance, and various real-world applications. A geometric sequence is defined by its initial term and a common ratio — the factor by which each term increases or decreases. This article explores one such sequence with specific values: the first term ( a = 3 ) and common ratio ( r = \frac{1.5}{3} = 0.5 ).", "## What is a Geometric Sequence?", "A geometric sequence is a series 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. This contrasts with arithmetic sequences, which use addition. With a well-defined ( a ) and ( r ), we gain predictable, exponential growth or decay.", "## The Given Values: ( a = 3 ) and ( r = 0.5 )", "- First term (( a )): The sequence starts at 3. So, the first few terms are directly determined by repeated multiplication by ( r ).\n- Common ratio (( r )): Calculated as ( \frac{1.5}{3} = 0.5 ), meaning each term is half of the previous term. This value makes the sequence a classic example of geometric decay—values shrinking toward zero.", "## Formulating the General Term", "The ( n^{\ ext{th}} ) term of a geometric sequence is given by:", "[\na_n = a \cdot r^{n-1}\n]", "Substituting ( a = 3 ) and ( r = 0.5 ):", "[\na_n = 3 \cdot (0.5)^{n-1}\n]", "This formula lets us compute any term:", "- ( a_1 = 3 \cdot (0.5)^0 = 3 )\n- ( a_2 = 3 \cdot (0.5)^1 = 1.5 )\n- ( a_3 = 3 \cdot (0.5)^2 = 0.75 )\n- ( a_4 = 3 \cdot (0.5)^3 = 0.375 ), and so on.", "The terms form the sequence: ( 3,\ 1.5,\ 0.75,\ 0.375,\ \dots )", "## Key Properties of This Geometric Sequence", "- Type of Sequence: Decreasing geometric sequence (since ( |r| < 1 ))\n- Behavior: It converges to zero as ( n \ o \infty ), because each term is half the prior one.\n- Exponential Decay: The multiplication by ( 0.5 ) drives rapid reduction — a fundamental property of geometric sequences with ( 0 < r < 1 ).", "## Real-World Applications", "Geometric sequences with small common ratios like ( 0.5 ) model phenomena involving exponential decay, such as:", "- Finance: Priors in discounted cash flows with consistent growth reductions.\n- Science: Radioactive decay rates where quantities halve over fixed periods.\n- Population Dynamics: Hypothetical scenarios of declining populations or decreasing contamination diffusion.", "Even though ( r = 0.5 ) represents a 50% reduction per stage, the principle applies broadly to any multiplicative decay process.", "## Summing the Sequence", "While the individual terms shrink, summing the infinite series reveals a closed-form sum:", "[\nS_{\infty} = \frac{a}{1 - r} = \frac{3}{1 - 0.5} = \frac{3}{0.5} = 6\n]", "This means the total sum of all terms from ( n = 1 ) to infinity converges to 6. This sum reflects the cumulative effect of repeated halving — an intuitive illustration of convergence in infinite geometric series.", "## Practical Uses and Calculations", "Understanding geometric sequences aids in:", "- Projecting future values in consistent decay scenarios.\n- Analyzing iterative algorithms involving multiplicative scaling.\n- Modeling compound interest with periodic reductions.", "Calculating specific terms or sums quickly using the formula ( a_n = 3 \cdot (0.5)^{n-1} ) streamlines computations in academic and applied contexts.", "## Conclusion", "The geometric sequence with ( a = 3 ) and ( r = 0.5 ) exemplifies clear exponential decay dynamics. Its predictable term progression, convergent sum, and real-world applicability make it a foundational example in math education and applied analysis. Whether in finance, science, or learning exponential functions, recognizing such sequences empowers deeper mathematical insight.", "---", "Keywords: geometric sequence, common ratio ( r = 0.5 ), exponential decay, real-world applications, infinite series sum ( \frac{a}{1 - r} ), first term ( a = 3 ), mathematical modeling.", "Meta Description: Dive into the geometric sequence with first term ( a = 3 ) and common ratio ( r = 0.5 ). Learn how its exponential decay shapes terms, sums, and real-life applications in finance, science, and tech."]