Problem 103 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I modeled California's population growth with a geometric sequence, so my model is an exponential function whose domain is the set of natural numbers.

Step-by-Step Solution

Answer

The statement makes sense. California's population growth can indeed be modeled using a geometric sequence, which can be represented as an exponential function. The domain being the set of natural numbers also makes sense, as the unit of growth is considered in whole numbers (years).

Step 1: Understanding the statement

The statement is saying that California's population growth can be modeled with a geometric sequence, and thus, it can be represented by an exponential function. A geometric sequence can indeed represent population growth if the population is increasing by a constant percentage or multiplying factor. In this case, the exponential function would hold the set of natural numbers. Natural numbers (counting numbers) make sense as the domain when we talk about population, as we can't have a fraction or a negative number of people.

Step 2: Correlation between geometric sequence and exponential function

A geometric sequence has a constant ratio between terms, whereas an exponential function increases or decreases at a constant rate. Therefore, a geometric sequence can also be represented through an exponential function. If the statement reflects a model where the population growth of California increases by a fixed percentage each year (geometric growth), it can be represented as an exponential function. The domain would indeed be the set of natural numbers, as we usually measure population growth in whole years.

Step 3: Final evaluation

After the examination of the statement, recalling the nature of geometric sequences and exponential functions, and considering the domain, it's clear that the statement makes sense. A model based on a geometric sequence for a population growth is equivalent to an exponential function where the domain is the set of natural numbers, considering years as the counted units.