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Question Write an explicit formula for \( a_{n} \), the \( n^{\text {th }} \) term of the sequence \( 8,-4,2, \ldots \) Answer Attempt 1 out of 3 \( a_{n}=\square \) Submit Answer

Ask by Mcguire Todd. in the United States
Mar 16,2025

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The explicit formula for the \( n^{\text{th}} \) term of the sequence is: \[ a_n = 8 \left(-\frac{1}{2}\right)^{(n-1)} \]

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The Deep Dive

To find the explicit formula for the sequence \(8, -4, 2, \ldots\), we can observe how the terms are changing. Each term is being multiplied by \(-\frac{1}{2}\) from the previous term. This gives the formula: \[ a_{n} = 8 \left(-\frac{1}{2}\right)^{n-1} \] This means that the first term \(a_1\) is \(8\), the second term \(a_2\) is \(-4\), and for \(n = 3\), \(a_3\) is \(2\). The sequence decreases in magnitude while alternating in sign due to the negative fraction being raised to increasing powers. For those who enjoy patterns, this sequence is an excellent example of geometric sequences where each term is derived from the previous one by a constant ratio. You can visualize it like a roller coaster, going up and down as you flip between positive and negative values. Dive deeper into the intricacies of sequences and you’ll find joy in their predictability! Have you ever noticed how the decay of these terms can be seen in real-world processes, like radioactive decay or depreciation of assets? The same principle applies! As values get smaller, we can mathematically model them using similar geometric sequences. This makes understanding such behaviors not only crucial in math but also in finance, biology, and many other fields!

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