Answer to Query "Will ALL arithmetic sequences be linear functions?"
Let's look at the formulas for an arithmetic function and for a linear function.
Arithmetic Sequence:
Linear Function:f (x) = mx + b
Arithmetic Function:
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.
Let's convert the Arithmetic Function to 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)
Let's 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.
