Question

how does the factorial function compare to the exponential function 2^n in terms of growth?
a. it grows at the same rate.
b. it decreases over time.
c. it grows faster.
d. it grows slower.

Explanation:

Step1: Analyze growth rates

For \(n = 1\), \(n! = 1\), \(2^n=2\). For \(n = 2\), \(n! = 2\), \(2^n = 4\). For \(n=3\), \(n! = 6\), \(2^n=8\). For \(n = 4\), \(n! = 24\), \(2^n = 16\). For \(n=5\), \(n! = 120\), \(2^n=32\).

Step2: Compare trends

As \(n\) increases, \(n!\) grows faster than \(2^n\). For example, using the ratio \(\frac{n!}{2^n}\), when \(n = 1\), \(\frac{1!}{2^1}=\frac{1}{2}\); when \(n=2\), \(\frac{2!}{2^2}=\frac{2}{4}=\frac{1}{2}\); when \(n = 3\), \(\frac{3!}{2^3}=\frac{6}{8}=\frac{3}{4}\); when \(n=4\), \(\frac{4!}{2^4}=\frac{24}{16}=\frac{3}{2}\); when \(n = 5\), \(\frac{5!}{2^5}=\frac{120}{32}=\frac{15}{4}\). As \(n\) gets larger, the ratio \(\frac{n!}{2^n}\) increases without bound.

Answer:

C. It grows faster.