Write an equation to describe the sequence below, use n to represent the position of a term in the sequence, where n = 1 for the first term: 3, 12, 48, 5, 25, ... Write your answer... Write an equation to describe the sequence below, use n to represent the position of a term in the sequence, where n = 1 for the first term: 3, 12, 48, 5, 25, ... Write your answer using proper fractions, improper fractions, and integers.
Understand the Problem
The question is asking for an equation that describes a geometric sequence given specific terms. The task involves identifying the pattern of the sequence and formulating a general expression based on the position of the term represented by n.
Answer
$$ a_n = 3 \cdot 4^{(n - 1)} $$
Answer for screen readers
$$ a_n = 3 \cdot 4^{(n - 1)} $$
Steps to Solve
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Identify the First Term The first term of the sequence is $a_1 = 3$.
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Identify the Common Ratio To find the common ratio, divide the second term by the first term: $$ r = \frac{a_2}{a_1} = \frac{12}{3} = 4 $$
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Write the General Formula The general formula for a geometric sequence can be written as: $$ a_n = a_1 \cdot r^{(n - 1)} $$
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Substitute Known Values Now substitute the values into the formula: $$ a_n = 3 \cdot 4^{(n - 1)} $$
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Final Answer Format Write the expression clearly: $$ a_n = 3 \cdot 4^{(n - 1)} $$
$$ a_n = 3 \cdot 4^{(n - 1)} $$
More Information
This expression represents the $n^{th}$ term of the geometric sequence where each term is four times the previous term, starting from 3.
Tips
- Forgetting to subtract 1 from $n$ in the exponent. Remember, the first term corresponds to $n = 1$.
- Not identifying the correct common ratio (dividing the wrong terms).
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