Write an equation to describe the sequence 3/2, 2, 8/3, ... Use n for the term position, where n = 1 for the first term. Write your answer using proper fractions, improper fraction... Write an equation to describe the sequence 3/2, 2, 8/3, ... Use n for the term position, where n = 1 for the first term. Write your answer using proper fractions, improper fractions, and integers.
Understand the Problem
The question asks to write an equation that describes a geometric sequence using the variable n to indicate the position of the terms. The user needs to express the nth term of the sequence in the form of a formula involving fractions and integers.
Answer
$$ a_n = \frac{4^{(n-1)}}{2 \cdot 3^{(n-1)}} $$
Answer for screen readers
$$ a_n = \frac{4^{(n-1)}}{2 \cdot 3^{(n-1)}} $$
Steps to Solve
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Identify the first term and common ratio The first term of the geometric sequence is given as $\frac{3}{2}$. To find the common ratio, divide the second term by the first: $$ r = \frac{2}{\frac{3}{2}} = \frac{2 \times 2}{3} = \frac{4}{3} $$
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General formula for the nth term The nth term of a geometric sequence can be expressed as: $$ a_n = a_1 \cdot r^{(n-1)} $$ where $a_1$ is the first term and $r$ is the common ratio.
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Substituting known values Substitute the values of $a_1$ and $r$ into the formula: $$ a_n = \frac{3}{2} \cdot \left(\frac{4}{3}\right)^{(n-1)} $$
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Simplifying the expression You can rewrite it as a single fraction: $$ a_n = \frac{3}{2} \cdot \frac{4^{(n-1)}}{3^{(n-1)}} = \frac{3 \cdot 4^{(n-1)}}{2 \cdot 3^{(n-1)}} = \frac{4^{(n-1)}}{2^{(n-1)} \cdot 3^{(n-1)}} $$
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Final form So the final formula for the nth term of the sequence is: $$ a_n = \frac{4^{(n-1)}}{2 \cdot 3^{(n-1)}} $$
$$ a_n = \frac{4^{(n-1)}}{2 \cdot 3^{(n-1)}} $$
More Information
This expression gives you the nth term of the geometric sequence defined by the initial terms $\frac{3}{2}, 2, 8, ...$. You can directly substitute different values of (n) to find specific terms in the sequence.
Tips
- Confusing the positions of the terms: Make sure you use the correct term for (n=1) and the appropriate exponent for the common ratio.
- Forgetting to simplify the fraction fully. Ensure you reduce it to its simplest form.
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