For the 6th term: \(a_6 = 3 \cdot 2^{(6-1)} = 3 \cdot 2^5 = 3 \cdot 32 = 96\).
["# Understanding the 6th Term in a Geometric Sequence: ( a_6 = 96 )", "Learning how to compute terms in a geometric sequence is a valuable math skill, especially when exploring exponential growth patterns. In this article, we’ll break down the formula and calculation for the 6th term of a specific geometric sequence:\n[ a_6 = 3 \cdot 2^{(6-1)} = 3 \cdot 2^5 = 3 \cdot 32 = 96 ]", "## What Is a Geometric Sequence?", "A geometric sequence is a series of numbers where each term is found by multiplying the previous term by a fixed positive number called the common ratio. Commonly, sequences like this follow the general form:\n[ a_n = a_1 \cdot r^{(n-1)} ]\nwhere:\n- ( a_n ) is the nth term,\n- ( a_1 ) is the first term,\n- ( r ) is the common ratio,\n- ( n ) is the term number.", "## Applying the Formula to Find the 6th Term", "Given the expression for the 6th term:\n[ a_6 = 3 \cdot 2^{(6-1)} ]\nwe identify:\n- Initial term ( a_1 = 3 ),\n- Common ratio ( r = 2 ),\n- Term number ( n = 6 ).", "Substitute into the formula:\n[\na_6 = 3 \cdot 2^{(6-1)} = 3 \cdot 2^5\n]\nNow calculate ( 2^5 ):\n[\n2^5 = 32\n]\nThen multiply:\n[\na_6 = 3 \cdot 32 = 96\n]", "## Why Is This Formula Useful?", "This formula efficiently finds any term in a geometric sequence without listing all preceding terms. It highlights exponential growth behavior—key in fields such as finance (compound interest), biology (bacterial growth), and computer science (algorithmic scaling).", "## Step-by-Step Summary", "| Step | Explanation | Calculation |\n|------|-----------------------------------------|----------------------------|\n| 1 | Use the geometric sequence formula | ( a_n = a_1 \cdot r^{(n-1)} ) |\n| 2 | Identify given values | ( a_1 = 3 ), ( r = 2 ), ( n = 6 ) |\n| 3 | Substitute into formula | ( a_6 = 3 \cdot 2^{5} ) |\n| 4 | Evaluate exponent | ( 2^5 = 32 ) |\n| 5 | Multiply by initial term | ( 3 \cdot 32 = 96 ) |", "## Conclusion", "Understanding how to calculate the 6th term of a geometric sequence—using the formula ( a_n = a_1 \cdot r^{(n-1)} )—enables quick and accurate solutions in diverse real-world applications. Remembering that for this case ( a_6 = 3 \cdot 2^5 = 96 ) shows how exponential patterns drive rapid growth, making geometric sequences a powerful concept in mathematics.", "Whether you're studying algebra, preparing for exams, or modeling real-life scenarios, mastering this formula empowers deeper mathematical insight and problem-solving confidence.", "---", "Keywords: geometric sequence, 6th term, exponential growth, ( a_6 = 3 \cdot 2^5 ), math formula, algebra tutorial, exponential calculation, sequence term formula."]
