To find the arithmetic sequence function that represents the given points \((0, 13)\), \((1, 20)\), and \((2, 27)\), we can first observe the outputs (y-values) of the points:
- For \(x = 0\), \(y = 13\)
- For \(x = 1\), \(y = 20\)
- For \(x = 2\), \(y = 27\)
Next, we can check the differences between successive y-values to confirm that this is indeed an arithmetic sequence:
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The difference between \(y\) values at \(x = 1\) and \(x = 0\) is: \[ 20 - 13 = 7 \]
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The difference between \(y\) values at \(x = 2\) and \(x = 1\) is: \[ 27 - 20 = 7 \]
Since the difference is constant (\(7\)), we can establish that this represents an arithmetic sequence.
To express this as a function, \(y\) can be represented in terms of \(x\) in the form of:
\[ y = mx + b \]
where \(m\) is the common difference and \(b\) is the y-intercept. From our earlier calculation, we found \(m = 7\).
Now we need to find the y-intercept \(b\). We know that when \(x = 0\), \(y = 13\), therefore:
\[ y = 7(0) + b \implies b = 13 \]
Putting this together, we get the function:
\[ y = 7x + 13 \]
Thus, the function representing the arithmetic sequence for the given points is:
\[ \boxed{y = 7x + 13} \]
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