To find the explicit formula for the arithmetic sequence given: -5, 13, 31, 49, ..., we first need to identify the first term and the common difference.
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Identify the first term (\(b(1)\)): The first term (\(b(1)\)) is -5.
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Calculate the common difference (\(d\)): To find the common difference, subtract the first term from the second term: \[ d = 13 - (-5) = 13 + 5 = 18 \] We can verify this by calculating the difference between other consecutive terms: \[ 31 - 13 = 18 \quad \text{and} \quad 49 - 31 = 18 \] Thus, the common difference is \(d = 18\).
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Write the explicit formula: The general formula for an arithmetic sequence can be expressed as: \[ b(n) = b(1) + (n - 1) \cdot d \] Substituting in our values: \[ b(n) = -5 + (n - 1) \cdot 18 \]
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Simplify the formula: Distributing the \(18\): \[ b(n) = -5 + 18n - 18 \] Combining like terms: \[ b(n) = 18n - 23 \]
Therefore, the explicit formula for the arithmetic sequence is: \[ \boxed{b(n) = 18n - 23} \]