A1Side
Query: Are Arithmetic Sequences
Linear Functions?

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"Will arithmetic sequences be linear functions?"

Let's compare the formulas for an arithmetic sequence with that of a linear function.
We will be using functional notation for the sequence.


Arithmetic Sequence:
    funcformulaF

Linear Function:    f (x) = mx + b

Arithmetic Sequence:
n
is the variable.
d is the rate of change
f (1) is a constant.

Linear Function:
x
is the variable.
m is the rate of change
b is a constant.


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Convert the Arithmetic Function to a Linear Function:

Rename the variable to x, and change the rate of change to m.
f (n) = f (1) + d (n - 1)
f (x) = f (1) + m (x - 1)

Distribute:    f (x) = f (1) + mx - m

Rearrange terms:    f (x) = f (1) - m + mx

Notice that f (1) - m is a constant term (a number).

Replace f (1) - m with b.
f (x) = b + mx

And we have:     f (x) = mx + b

Since f (x) = y, we can also write y = mx + b.

Arithmetic sequences are linear functions.

 


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