Compare the linear function and the arithmetic sequence below.
Linear Function:
x y
-1 10
0 14
1 18
2 22
The rate of change of the linear function is
4
The y-intercept of the linear function is
10
Arithmetic Sequence (Explicit formula):
an = 14 + 4(n−1)
The rate of change (common difference) of the arithmetic sequence is
14
The initial value of the arithmetic sequence is
4
The rate of change of this linear function is
equal to
the rate of change of this arithmetic sequence.
Arithmetic sequences
always
can be represented as a linear function.
Both the linear function and the arithmetic sequence have a rate of change, also known as a slope. In the case of the linear function, the rate of change is constant and equal to 4. In the arithmetic sequence, the rate of change is also constant and equal to 14.
Additionally, the linear function has a y-intercept, which is the value of y when x is 0. In this case, the y-intercept is 10. The arithmetic sequence does not have a y-intercept because it is an ordered list of numbers rather than a graph.
The explicit formula for the arithmetic sequence shows that each term in the sequence is found by adding 4 to the previous term. This is similar to the linear function where the value of y is found by adding 4 to the value of x.
Overall, the rate of change and the concept of adding a constant value are similarities between the linear function and the arithmetic sequence. However, the linear function can be represented graphically, while the arithmetic sequence is represented as a list of numbers.