To determine the explicit formula for the sequence \( { a_n } = {-3, -\frac{1}{2}, 2, \frac{9}{2}, 7, \ldots} \), we first examine the pattern in the sequence values.
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Finding the First Differences: Let's calculate the differences between consecutive terms:
- \( -\frac{1}{2} - (-3) = 2.5 \)
- \( 2 - (-\frac{1}{2}) = 2.5 \)
- \( \frac{9}{2} - 2 = 2.5 \)
- \( 7 - \frac{9}{2} = 2.5 \)
The first differences are constant (all equal to \( 2.5 \)), indicating that the sequence is linear.
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Finding the General Form: A linear sequence can be expressed in the form: \[ a_n = mn + b \] where \( m \) is the common difference and \( b \) is the first term of the sequence adjusted based on the index.
We already found that the common difference \( m = \frac{5}{2} \) because the difference is \( 2.5 \).
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Finding the Explicit Formula: We can find \( b \) using the first term:
- At \( n = 1 \), \( a_1 = -3 \): \[ a_1 = m \cdot 1 + b = \frac{5}{2} \cdot 1 + b = -3 \implies b = -3 - \frac{5}{2} = -\frac{6}{2} - \frac{5}{2} = -\frac{11}{2} \]
Now, our explicit formula for the sequence is: \[ a_n = \frac{5}{2}n - \frac{11}{2} \]
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Finding \( a_{17} \): Now, we calculate \( a_{17} \): \[ a_{17} = \frac{5}{2} \cdot 17 - \frac{11}{2} = \frac{85}{2} - \frac{11}{2} = \frac{74}{2} = 37 \]
Therefore, the explicit formula is: \[ a_n = \frac{5}{2}n - \frac{11}{2} \] and the value of \( a_{17} \) is: \[ \boxed{37} \]
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