A geometric sequence starts with 5 and has a common ratio of 3. What is the 4th term?
["Understanding Geometric Sequences: Find the 4th Term When Starting with 5 and a Common Ratio of 3", "A geometric sequence is a fascinating mathematical pattern where each term is generated by multiplying the previous term by a constant value known as the common ratio. This elegant structure shows up in nature, finance, and science—and knowing how to compute its terms makes it easier to model real-world growth and change.", "### What is a Geometric Sequence?", "In a geometric sequence, the first term is called ( a ), and each subsequent term is obtained by multiplying the previous term by the common ratio ( r ). The general formula for the ( n )-th term ( a_n ) is:", "[\na_n = a \ imes r^{n-1}\n]", "### Given Example: Start with 5, Common Ratio = 3", "Here, the first term ( a = 5 ) and the common ratio ( r = 3 ). This means each term is 3 times the one before it.", "Let’s calculate the first few terms to see the pattern clearly:", "- 1st term: ( a_1 = 5 \ imes 3^{1-1} = 5 \ imes 3^0 = 5 \ imes 1 = 5 )\n- 2nd term: ( a_2 = 5 \ imes 3^{2-1} = 5 \ imes 3^1 = 15 )\n- 3rd term: ( a_3 = 5 \ imes 3^{3-1} = 5 \ imes 3^2 = 45 )\n- 4th term: ( a_4 = 5 \ imes 3^{4-1} = 5 \ imes 3^3 = 5 \ imes 27 = 135 )", "### Why Is the 4th Term 135?", "Using the formula:", "[\na_4 = 5 \ imes 3^{4-1} = 5 \ imes 3^3 = 5 \ imes 27 = 135\n]", "So, the 4th term is 135.", "### Summary", "- The first term is 5\n- The common ratio is 3\n- Using the geometric sequence formula, the 4th term is ( 5 \ imes 3^{3} = 135 )", "Understanding geometric sequences helps in predicting growth patterns—whether in population studies, compound interest, or data modeling. With a starting value of 5 and a multiplicative climb of 3, the 4th term stands firmly at 135.", "---", "Key Lightbulb Moment: Geometric sequences are everywhere. Once you recognize the pattern, forecasting the future terms becomes simple and powerful!", "Keywords:** geometric sequence, geometric progression, common ratio, formula for nth term, 4th term geometric sequence, math tutorial, algebra, sequence calculation."]
