To find an explicit formula for the given arithmetic sequence \(8, 1, -6, -13\), we need to determine the first term and the common difference.
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Identify the first term (\(a_1\)): The first term \(a_1\) is \(8\).
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Calculate the common difference (\(d\)): The common difference is found by subtracting the first term from the second term: \[ d = a_2 - a_1 = 1 - 8 = -7 \]
We can check the common difference using the next pairs of terms: \[ d = a_3 - a_2 = -6 - 1 = -7 \] \[ d = a_4 - a_3 = -13 - (-6) = -13 + 6 = -7 \] The common difference \(d = -7\) is consistent throughout the sequence.
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Write the explicit formula: The general explicit formula for the \(n\)-th term of an arithmetic sequence can be written as: \[ a_n = a_1 + (n-1)d \]
Substituting the values we found: \[ a_n = 8 + (n-1)(-7) \]
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Simplify the formula: \[ a_n = 8 - 7(n - 1) \] \[ a_n = 8 - 7n + 7 \] \[ a_n = 15 - 7n \]
Thus, the explicit formula for the arithmetic sequence is: \[ \boxed{a_n = 15 - 7n} \]