Question

Can all linear functions be modeled by an arithmetic sequence? if not, provide a counterexample

Answer 1

It depends. Generally no.

Linear equations are generally in the form [math]y=mx+b[/math] and have a domain of [math](-\infty,\infty)[/math], or all real numbers. However, an arithmetic sequence is only defines for the natural numbers (that is, while numbers [math]> 0[/math].

For any two terms in the arithmetic sequence, [math]a_n[/math] and [math]a_{n+1}[/math], there will always be a point on the linear function that lies in between them, and is such not defined in the sequence.

This does not make the sequence and function unrelated, but rather it makes them not the same.

A similar argument applies for geometric sequences and exponential equations.