Final answer:
The sequence 12, 144, 1728 is a geometric sequence because each term is 12 times the previous term, displaying exponential growth.
Step-by-step explanation:
To determine whether the given sequence, 12, 144, 1728, is arithmetic or geometric, let's examine the pattern of change between the terms. In an arithmetic sequence, a constant is added to go from one term to the next, whereas in a geometric sequence, each term is multiplied by a constant to get to the next term.
In this sequence, if we divide each term by its predecessor:
- 144 ÷ 12 = 12
- 1728 ÷ 144 = 12
We see that each term is indeed 12 times the previous term, indicating that we have a consistent multiplicative factor. This is a classic characteristic of
exponential growth
where the growth rate is constant in terms of multiplication. Thus, the correct identification for this sequence is
geometric
; it is characterized by exponential growth by a factor of 12.