which statements correctly describe how the graph of the geometric sequence below should appear? 640, 160…

Explanation:

Step1: Identify the common - ratio

For a geometric sequence (a_n), the common - ratio (r=\frac{a_{n + 1}}{a_n}). Here, (a_1 = 640), (a_2=160), so (r=\frac{160}{640}=\frac{1}{4}). Since (0\lt r\lt1), the sequence is a decaying sequence.

Step2: Analyze the domain and range

The terms of a sequence are numbered as the first term ((n = 1)), the second term ((n = 2)), etc. So the domain of a sequence is the set of natural numbers ({1,2,3,\cdots}). The terms of the given sequence are (640,160,40,10,\cdots) which are natural numbers, but the sequence will continue to get smaller and approach (0) but never reach (0). The range is a subset of the natural numbers. Also, the general form of a geometric sequence is (a_n=a_1r^{n - 1}), which is an exponential function of (n). When (0\lt r\lt1), it represents exponential decay.

Answer:

C. The domain will be the set of natural numbers. E. The graph will show exponential decay.