A geometric sequence starts with 2 and has a common ratio of 3. What is the 6th term?
["Understanding Geometric Sequences: Finding the 6th Term Starting with 2 and a Common Ratio of 3", "A geometric sequence is a powerful mathematical concept used in various fields such as finance, biology, and computer science. If you're learning about sequences or tackling a problem involving exponential growth, understanding how geometric sequences work is essential. In this article, we explore a classic example: a geometric sequence that begins with 2 and has a common ratio of 3, and we’ll calculate its 6th term step by step.", "### What Makes a Sequence a Geometric Sequence?", "A geometric sequence is defined by a starting term (called the first term) and a constant ratio by which each term is multiplied to get the next term. Formally, if ( a_1 ) is the first term and ( r ) is the common ratio, then each term is given by:", "[\na_n = a_1 \ imes r^{(n-1)}\n]", "This formula lets you find any term in the sequence without listing all previous terms.", "### Given Values in Our Problem", "In this specific example:", "- First term (( a_1 )) = 2\n- Common ratio (( r )) = 3\n- We want the 6th term (( a_6 ))", "### Calculating the 6th Term Using the Formula", "Plug the known values into the formula:", "[\na_6 = a_1 \ imes r^{(6-1)} = 2 \ imes 3^5\n]", "Now calculate ( 3^5 ):", "[\n3^5 = 3 \ imes 3 \ imes 3 \ imes 3 \ imes 3 = 243\n]", "Then multiply by the first term:", "[\na_6 = 2 \ imes 243 = 486\n]", "### Conclusion", "The 6th term in the geometric sequence starting with 2 and having a common ratio of 3 is 486. This demonstrates how quickly values grow in geometric sequences due to repeated multiplication, a phenomenon seen often in compound interest, population growth, and fractal patterns.", "Understanding these sequences equips learners with tools to model real-world exponential changes effectively.", "Keywords: geometric sequence, common ratio, sixth term calculation, exponential growth, mathematics education, common ratio example, sequence formula."]
