To find the explicit formula for the given arithmetic sequence \(-12, -20, -28, -36\), we first need to identify the first term and the common difference.
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First term (\(a_1\)): The first term of the sequence is \(a_1 = -12\).
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Common difference (\(d\)): To find the common difference, we can subtract the first term from the second term: \[ d = a_2 - a_1 = -20 - (-12) = -20 + 12 = -8 \] We can verify this by checking the difference between other consecutive terms: \[ -28 - (-20) = -28 + 20 = -8 \ -36 - (-28) = -36 + 28 = -8 \] The common difference \(d\) is consistently \(-8\).
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Explicit formula: The general formula for the \(n\)-th term of an arithmetic sequence can be expressed as: \[ a_n = a_1 + (n - 1) d \] Substituting the known values \(a_1 = -12\) and \(d = -8\) into the formula: \[ a_n = -12 + (n - 1)(-8) \] Simplifying the formula: \[ a_n = -12 - 8(n - 1) \ = -12 - 8n + 8 \ = -8n - 4 \]
Thus, the explicit formula for the arithmetic sequence is: \[ a_n = -8n - 4 \]