If a geometric sequence has a first term of 5 and a common ratio of 3, what is the 4th term?
["# Understanding Geometric Sequences: Finding the 4th Term with a First Term of 5 and a Common Ratio of 3", "When exploring mathematical patterns, geometric sequences stand out for their clear and predictable structure. If you’ve ever wondered how to calculate a specific term in a geometric sequence, you’re in the right place. In this article, we’ll explore what a geometric sequence is, how to identify its key components, and solve a classic problem: Finding the 4th term when the first term is 5 and the common ratio is 3.", "## What is a Geometric Sequence?", "A geometric sequence is a list 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 pattern is represented mathematically as:", "[\na_n = a_1 \cdot r^{n-1}\n]", "Where:\n- ( a_n ) = the ( n^\ ext{th} ) term,\n- ( a_1 ) = the first term,\n- ( r ) = the common ratio,\n- ( n ) = the term number.", "## Given Values in the Problem", "In our specific example:", "- The first term ( a_1 = 5 )\n- The common ratio ( r = 3 )\n- We want to find the 4th term, so ( n = 4 )", "## Step-by-Step Calculation for the 4th Term", "Using the formula:", "[\na_4 = a_1 \cdot r^{4-1} = a_1 \cdot r^3\n]", "Substitute the given values:", "[\na_4 = 5 \cdot 3^3\n]", "Calculate the exponent:", "[\n3^3 = 27\n]", "Now multiply:", "[\na_4 = 5 \cdot 27 = 135\n]", "## Conclusion", "Therefore, the 4th term in the geometric sequence with a first term of 5 and a common ratio of 3 is 135.", "Understanding this formula not only solves the immediate question but also empowers you to analyze and predict behavior in broader contexts—whether in finance, population growth, or fractal patterns—where geometric sequences frequently apply.", "---", "Keywords for SEO: geometric sequence, 4th term calculation, first term 5, common ratio 3, mathematical sequence formula, exponential growth in sequences, solve geometric sequence problem", "meta description: Learn how to find the 4th term of a geometric sequence with first term 5 and common ratio 3 using the formula ( a_n = a_1 \cdot r^{n-1} ). Step-by-step solution and explanation included."]
