4th term: \( a_4 = 5 \times 3^{3} = 5 \times 27 = 135 \)

["Understanding the 4th Term of the Geometric Sequence: ( a_4 = 5 \ imes 3^{3} = 135 )", "When exploring sequences in algebra, one of the most fascinating concepts is geometric sequences—where each term grows by a constant ratio. A key aspect of working with such sequences is evaluating any specific term, especially the 4th term, using the general term formula.", "—", "## What is the 4th Term in a Geometric Sequence?", "In a geometric sequence, the general term is defined as:\n[\na_n = a_1 \ imes r^{n-1}\n]\nwhere:\n- ( a_n ) is the ( n )-th term,\n- ( a_1 ) is the first term,\n- ( r ) is the common ratio,\n- ( n ) is the term number.", "The 4th term (( a_4 )) follows directly from this formula:\n[\na_4 = a_1 \ imes r^{4-1} = a_1 \ imes r^3\n]", "---", "### Solving for ( a_4 = 5 \ imes 3^3 )", "In this example, the expression ( a_4 = 5 \ imes 3^3 ) shows that:\n- The first term ( a_1 = 5 ),\n- The common ratio ( r = 3 ),\n- Therefore, ( a_4 = 5 \ imes 3^3 = 5 \ imes 27 = 135 ).", "Let’s break it down step-by-step:", "1. Substitute values into the general term:\n[\na_4 = 5 \ imes 3^{4-1} = 5 \ imes 3^3\n]", "2. Calculate the exponent:\n[\n3^3 = 27\n]", "3. Multiply:\n[\na_4 = 5 \ imes 27 = 135\n]", "This confirms that the 4th term in this geometric sequence is 135.", "---", "### Why This Matters", "Understanding how to derive specific terms like ( a_4 ) using exponent rules enhances your ability to solve complex algebraic problems, financial models involving compound interest, or scientific sequences like population growth or decay calculations.", "---", "### Quick Recap:\n- Geometric sequences grow by a constant ratio: ( a_n = a_1 \ imes r^{n-1} )\n- For ( a_4 ), the exponent is 3\n- ( 3^3 = 27 ), so ( 5 \ imes 27 = 135 )\n- This method empowers efficient term evaluation in recurring exponential patterns", "For more insights into sequences, solving equations, and pattern recognition, explore algebraic tools and resources tailored for students and lifelong learners.", "---", "Keywords: geometric sequence, 4th term formula, ( a_4 = 5 \ imes 3^3 ), exponential terms, algebra learning, sequence evaluation, mathematical growth patterns", "---", "True appreciation of mathematics begins with mastering foundational patterns—like the 4th term of a geometric sequence. Keep calculating, keep growing!"]