To determine whether the points (0, 14), (1, 20), and (2, 27) form an arithmetic sequence, we first need to check the differences between the successive \( y \)-values.
-
The difference between the second point and the first point: \[ 20 - 14 = 6 \]
-
The difference between the third point and the second point: \[ 27 - 20 = 7 \]
Since the differences (6 and 7) are not constant, the \( y \)-values do not form an arithmetic sequence.
If you are looking for a specific arithmetic sequence based on these points, we can deduce a linear equation through interpolation. A linear function has the form \( y = mx + b \), where \( m \) is the slope.
To find the slope (\( m \)) between the first and second points: \[ m = \frac{20 - 14}{1 - 0} = 6 \] To find the slope (\( m \)) between the second and third points: \[ m = \frac{27 - 20}{2 - 1} = 7 \]
Since the slopes are not the same, they cannot be represented by a single linear equation either. If you need an arithmetic sequence based on a rule or pattern, you can proceed by defining a new sequence starting from a common difference.
If you want an arithmetic sequence that fits between or around those pairs, here’s how you could construct it:
- Start from the first point \( (0, 14) \).
- Choose a common difference \( d = 6 \):
- The first term: \( 14 \)
- The second term: \( 14 + 6 = 20 \)
- The third term: \( 20 + 6 = 26 \)
Thus, a potential arithmetic sequence with \( d = 6 \) would be \( 14, 20, 26 \).
Or, if you want to consider \( d = 7 \):
- Start again from \( (0, 14) \).
- The first term: \( 14 \)
- The second term: \( 14 + 7 = 21 \)
- The third term: \( 21 + 7 = 28 \)
In this case, the arithmetic sequence would be \( 14, 21, 28 \).
In summary, while the given points do not form an arithmetic sequence themselves, you can construct various arithmetic sequences around them by selecting different common differences.
4 answers