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Tell whether the sequence is arithmetic. Explain your reasoning. \( 12,6,0,-6,-12 \ldots \) The sequence is \( \checkmark \) arithmetic because the

Ask by Fitzgerald Hobbs. in the United States
Mar 10,2025

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Answer

The sequence is arithmetic because the difference between consecutive terms is always \(-6\).

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The sequence is arithmetic. 1. The first term is \(a_1=12\). 2. The second term is \(a_2=6\). The difference between the second and first term is: \[ 6 - 12 = -6. \] 3. The third term is \(a_3=0\). The difference between the third and second term is: \[ 0 - 6 = -6. \] 4. The fourth term is \(a_4=-6\). The difference between the fourth and third term is: \[ -6 - 0 = -6. \] 5. Since the difference between consecutive terms is constant and equals \(-6\), the sequence is arithmetic.

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The sequence is \( \checkmark \) arithmetic because the difference between consecutive terms is constant. In this case, subtracting each term from the next reveals a common difference of -6. For example, \( 6 - 12 = -6 \), \( 0 - 6 = -6 \), and so forth. This consistent difference indicates that it qualifies as an arithmetic sequence. Additionally, recognizing arithmetic sequences can be beneficial in various real-life scenarios, such as budgeting or planning events. For instance, if you're saving a fixed amount of money every month, the total savings over time forms an arithmetic sequence. This understanding allows for effective financial planning, making it easier to reach future goals.

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