To derive an equation for the given sequence: \(3, 12, 48, ...\), we first observe how the sequence progresses.
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Let's list out the terms with their positions:
- \(a_1 = 3\)
- \(a_2 = 12\)
- \(a_3 = 48\)
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Next, let's examine the pattern:
- From \(a_1\) to \(a_2\): \(12 = 3 \times 4\)
- From \(a_2\) to \(a_3\): \(48 = 12 \times 4\)
We see a multiplication pattern. Each term seems to be multiplied by 4 to get the next term.
- To represent this as a formula, we can see that:
- The first term is \(3\).
- Each subsequent term is multiplied by \(4\) based on the previous term.
The \(n\)-th term of the sequence can be formulated as: \[ a_n = 3 \times 4^{n-1} \]
So the equation to describe the sequence is: \[ a_n = 3 \cdot 4^{n-1} \]
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