To find an explicit formula for the arithmetic sequence given, we first need to determine the first term and the common difference.
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Identify the first term (\(a_1\)):
- The first term of the sequence is \(a_1 = -12\).
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Calculate the common difference (\(d\)):
- The second term is \(a_2 = -\frac{35}{3}\).
- The common difference is calculated as follows: \[ d = a_2 - a_1 = -\frac{35}{3} - \left(-12\right) \]
- To subtract \(-12\), we convert it to a fraction with a denominator of 3: \[ -12 = -\frac{36}{3} \]
- Now we can calculate \(d\): \[ d = -\frac{35}{3} + \frac{36}{3} = \frac{1}{3} \]
Now we have \(a_1 = -12\) and \(d = \frac{1}{3}\).
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Write the explicit formula: The explicit formula for an arithmetic sequence is given by: \[ a_n = a_1 + (n-1)d \] Substituting the values of \(a_1\) and \(d\): \[ a_n = -12 + (n-1) \cdot \frac{1}{3} \]
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Simplify the formula: \[ a_n = -12 + \frac{n-1}{3} = -12 + \frac{n}{3} - \frac{1}{3} = -\frac{36}{3} - \frac{1}{3} + \frac{n}{3} = \frac{n - 37}{3} \]
Thus, the explicit formula for the arithmetic sequence is: \[ a_n = \frac{n - 37}{3} \]