A geometric sequence has a first term of 3 and a common ratio of 2. What is the 6th term in the sequence?
["Title: How to Find the 6th Term in a Geometric Sequence with First Term 3 and Common Ratio 2", "Understanding geometric sequences is essential in mathematics, especially in fields like finance, computer science, and algebra. A geometric sequence is defined by its first term and a constant ratio between consecutive terms. In this article, we’ll explore a specific geometric sequence — one with a first term of 3 and a common ratio of 2 — and determine the 6th term. Whether you’re a student, teacher, or Math enthusiast, this guide will clarify how geometric sequences work and how to solve for any term efficiently.", "### 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 makes geometric sequences vital for modeling exponential growth or decay, such as population growth, compound interest, and geometric shapes that grow proportionally.", "### The General Formula for the nth Term", "The nth term of a geometric sequence can be calculated using the formula:", "[\na_n = a_1 \cdot r^{(n-1)}\n]", "Where:\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", "This formula shows that once the first term and common ratio are known, every term in the sequence is easily computable.", "### Applying the Formula to the Given Sequence", "In this example, we are given:\n- First term (( a_1 )): 3\n- Common ratio (( r )): 2\n- Term to find: ( n = 6 )", "Plugging these values into the formula:", "[\na_6 = 3 \cdot 2^{(6-1)} = 3 \cdot 2^5\n]", "Now compute ( 2^5 ):", "[\n2^5 = 32\n]", "Then multiply by the first term:", "[\na_6 = 3 \cdot 32 = 96\n]", "### The 6th Term in the Sequence", "Therefore, the 6th term of the geometric sequence with ( a_1 = 3 ) and ( r = 2 ) is 96.", "### Verifying the Sequence", "To confirm, we can list the first six terms step-by-step:\n1st term: ( 3 )\n2nd term: ( 3 \cdot 2 = 6 )\n3rd term: ( 6 \cdot 2 = 12 )\n4th term: ( 12 \cdot 2 = 24 )\n5th term: ( 24 \cdot 2 = 48 )\n6th term: ( 48 \cdot 2 = 96 )", "This matches our calculation, confirming the 6th term is indeed 96.", "### Why Knowing the 6th Term Matters", "While this may seem like a simple tool, understanding how to locate specific terms in a geometric sequence builds a strong foundation for:\n- Modeling exponential datasets\n- Solving real-world problems involving growth or decay\n- Enhancing algebraic reasoning and pattern recognition", "### Final Thoughts", "Mastering geometric sequences empowers learners to recognize and apply exponential patterns in countless mathematical and scientific contexts. The formula for the nth term provides a quick and reliable way to find any position in the sequence—knowing that with first term 3 and ratio 2, the 6th term is 96. With practice, identifying sequences and calculating their terms becomes second nature, making math both accessible and powerful.", "---", "Keywords: geometric sequence, nth term formula, mathematical sequence, exponential growth, common ratio 2, first term 3, how to calculate geometric term, algebra practice problem."]
