Question

Consider the following situations that generate a sequence. a. Write out the first five terms of the sequence. b. Find an explicit formula for the terms of the sequence. c. Find a recurrence relation that generates the sequence. d. Using a calculator or a graphing utility, estimate the limit of the sequence or state that it does not exist. Population growth When a biologist begins a study, a colony of prairie dogs has a population of \(250 .\) Regular measurements reveal that each month the prairie dog population increases by \(3 \%\) Let \(p_{n}\) be the population (rounded to whole numbers) at the end of the \(n\) th month, where the initial population is \(p_{0}=250\).

Short answer

Question: Analyze the prairie dog population growth sequence, which increases by 3% each month, and find: a. The first five terms of the sequence. b. An explicit formula for the terms of the sequence. c. A recurrence relation that generates the sequence. d. Estimate the limit of the sequence if it exists, or state that it doesn't exist. Answer: a. The first five terms of the sequence are: 250, 257, 265, 273, and 281. b. The explicit formula for the terms of the sequence is: $$ p_n = 250(1 + 0.03)^n $$. c. The recurrence relation for the sequence is: $$ p_{n}=(1+0.03)p_{n-1} $$ for n > 0, with \( p_0 = 250 \). d. The limit of the sequence does not exist, as the prairie dog population will continually grow and not approach a fixed value.

Step-by-step solution

Understand the sequence and generate its first five terms.

According to the given situation, the prairie dog population increases by 3% each month. Let's generate the first five terms of the sequence. Term \(p_0\) (initial population): 250 Term \(p_1\): 250 + 3% of 250 = 250 + 0.03 * 250 = 250 + 7.5 = 257 (rounded to the nearest whole number) Term \(p_2\): 257 + 3% of 257 = 257 + 0.03 * 257 = 257 + 7.71 = 265 (rounded to the nearest whole number) Term \(p_3\): 265 + 3% of 265 = 265 + 0.03 * 265 = 265 + 7.95 = 273 (rounded to the nearest whole number) Term \(p_4\): 273 + 3% of 273 = 273 + 0.03 * 273 = 273 + 8.19 = 281 (rounded to the nearest whole number)

Find an explicit formula for the sequence.

To find an explicit formula for the prairie dog population at the end of the nth month, we need to incorporate the 3% monthly increase into the formula. The equation for exponential growth is $$ p_n = p_0 (1 + r)^n $$, where \(p_0\) is the initial population, \(r\) is the rate of growth, and \(n\) is the number of periods. In this case, \(p_0 = 250\) and \(r = 0.03\). So the explicit formula for the population at the end of the nth month is: $$ p_n = 250(1 + 0.03)^n $$

Find a recurrence relation for the sequence.

The recurrence relation can be found by considering the relationship between consecutive terms in the sequence. In this case, each term is 3% greater than the previous term. The general form for recurrence relation can be written as: $$ p_{n}=(1+0.03)p_{n-1} $$ for n > 0, with \( p_0 = 250 \).

Estimate the limit of the sequence or state that it doesn't exist.

Since the prairie dog population grows at a rate of 3% each month, it will continue to increase indefinitely (assuming no other factors limit the growth). Therefore, there is no limit to the sequence, as it will continually grow and not approach a fixed value. The limit of the sequence does not exist.

Key concepts

Explicit Formula for Sequences

An explicit formula for a sequence provides a direct way to calculate the value of any term based on its position in the sequence. This differs from recursive formulas, which require the previous terms to determine the next one.

Understanding explicit formulas is crucial for quickly identifying a term's value without iterating through the entire sequence. For a sequence representing population growth, an explicit formula often involves exponential functions, reflecting consistent percentage-based growth over time. The formula for the prairie dog population's nth month is a perfect example:
\[ p_n = p_0(1 + r)^n \]
Here, \( p_0 \) is the initial population, \( r \) represents the growth rate as a decimal, and \( n \) specifies the term number or the number of time periods.

Recurrence Relation

A recurrence relation is an equation that represents a sequence where each term is defined in terms of the previous terms. It is particularly helpful when modeling situations where the next step depends on the current state, such as population sequences.

In the context of our prairie dog population problem, the recurrence relation is:
\[ p_{n} = (1 + 0.03)p_{n-1} \]
which simply means that the population of the next month \( p_{n} \) is the previous month's population \( p_{n-1} \) increased by 3%. This relation is recursive because it requires knowing the population of the current month to determine the next.

Exponential Growth

Exponential growth occurs when a quantity increases at a rate proportional to its current value, resulting in the quantity growing faster and faster as it gets larger. This type of growth is common in populations, finance, and natural phenomena.

For the prairie dog colony, this growth is represented by a 3% increase each month. It leads to a rapid escalation in the number of prairie dogs as time progresses, barring any external limiting factors. The use of percent changes in a base value over time periods translates into the population following an exponential curve mathematically described by the formula:\[ p_n = p_0(1 + r)^n \]

Limits of Sequences

The limit of a sequence is the value that the terms of the sequence approach as the index (typically denoted as \( n \)) goes to infinity. If the terms grow unbounded or fluctuate infinitely without settling on a single value, the sequence is said to have no limit.

In cases of unbounded exponential growth, like our prairie dog population that increases by 3% each month, the population will theoretically grow to infinity. For the sequence:\[ p_n = 250(1 + 0.03)^n \]
as \( n \) approaches infinity, so does \( p_n \). Under these conditions, the limit does not exist. In practical scenarios, other factors such as resource limitations would eventually impede this growth.