To find the explicit formula for the sequence \(3, 7, 11, 15, 19, \ldots\), we first identify the pattern in the sequence.
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Identify the differences: \[ 7 - 3 = 4, \quad 11 - 7 = 4, \quad 15 - 11 = 4, \quad 19 - 15 = 4 \] The common difference is \(4\).
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Recognize the type of sequence: This is an arithmetic sequence where the first term \(a_1 = 3\) and the common difference \(d = 4\).
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Write the explicit formula: The general formula for the \(n\)-th term of an arithmetic sequence is given by: \[ a_n = a_1 + (n - 1)d \] Substituting \(a_1 = 3\) and \(d = 4\) into the formula: \[ a_n = 3 + (n - 1) \cdot 4 \] Simplifying this: \[ a_n = 3 + 4n - 4 = 4n - 1 \]
Thus, the explicit formula for the sequence is: \[ a_n = 4n - 1 \]
If you are looking for the correct subscript notation to represent this formula, it would be: \[ a_n = 4n - 1 \] Where \(n\) is the index of the term in the sequence starting from \(1\).
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