Recall that a geometric sequence can be described by the equation shown below.
a_n = a_1 r^(n-1)
In this format a_n is the nth term, a_1 is the initial term, and r is the common ratio. Now let's review the form of an exponential equation.
y=ab^x
Here a ≠0, b>0, and b≠1. If we compare both equations we can see that they have the same form. Note in particular that the common ratio r of a geometric sequence plays the role of the base b of an exponential function.
&Geometic Sequence &&Exponential function
& a_n = a_1 r^(n-1) && y= a b^()#FF8C00x
Therefore, if r>0 we can write an exponential function with b=r, and the points of the sequence would overlap the graph of the function.
On the other hand, if r<0 this is not possible, since by definition b cannot be negative for an exponential function. In a case like this, the graph of the sequence forms a pattern of points alternating between two different quadrants instead.
