To determine the function of the arithmetic sequence that corresponds to the points given, we need to identify the pattern in the coordinates provided: (0, 13), (1, 20), and (2, 27).
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Identify the change in y-values:
- From (0, 13) to (1, 20): The change in y is \(20 - 13 = 7\).
- From (1, 20) to (2, 27): The change in y is \(27 - 20 = 7\).
The consistent change in y-values is 7, indicating a constant difference, which is characteristic of an arithmetic sequence.
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Find the initial term and common difference:
- The first term (when x = 0) is \(y = 13\).
- The common difference \(d = 7\).
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Write the function: The formula for the nth term of an arithmetic sequence is given by: \[ a_n = a_1 + (n - 1) * d \] where:
- \(a_1\) is the first term,
- \(d\) is the common difference.
In terms of the coordinate plane where x represents \(n\) (the term number): \[ a_n = 13 + (n - 0) * 7 \] Simplifying this gives: \[ a_n = 13 + 7n \]
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Express as a function: Thus, the function of the graphed arithmetic sequence can be written as: \[ f(x) = 7x + 13 \]
Therefore, the function of the arithmetic sequence is \(f(x) = 7x + 13\).