To find the arithmetic sequence based on the given points (0, 13), (1, 20), and (2, 27), we first observe the values of the sequence at the respective positions:
- At position 0, the value is 13.
- At position 1, the value is 20.
- At position 2, the value is 27.
Next, we calculate the common difference \(d\) of the arithmetic sequence by finding the difference between consecutive terms:
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From 13 (position 0) to 20 (position 1): \[ d = 20 - 13 = 7 \]
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From 20 (position 1) to 27 (position 2): \[ d = 27 - 20 = 7 \]
Since the common difference \(d\) is the same (7), we confirm that this is indeed an arithmetic sequence.
The general formula for the \(n\)-th term of an arithmetic sequence can be expressed as: \[ a_n = a_0 + n \cdot d \] where \(a_0\) is the first term, and \(d\) is the common difference.
In this case:
- \(a_0 = 13\)
- \(d = 7\)
Thus, the \(n\)-th term of the sequence becomes: \[ a_n = 13 + 7n \]
Now, we can write out the first few terms of this arithmetic sequence:
- For \(n = 0\): \(a_0 = 13\)
- For \(n = 1\): \(a_1 = 20\)
- For \(n = 2\): \(a_2 = 27\)
- For \(n = 3\): \(a_3 = 34\) (which is \(13 + 7 \cdot 3\))
- For \(n = 4\): \(a_4 = 41\) (which is \(13 + 7 \cdot 4\))
So, the arithmetic sequence based on the points provided is: \[ 13, 20, 27, 34, 41, \ldots \]
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