3. The United States population grows at a rate of 0.5% a year. We have around 328.2 million. What will the population be in 50 years? Write the recursive and explicit formula. a. What are the first 3 terms? | b. Recursive: 9₁ = an c. Explicit: 9₁ = f(n) = d. What will the population be in 50 years? Remember, we have to find the 51st term to find the 50th year. f( )=
3. The United States population grows at a rate of 0.5% a year. We have around 328.2 million. What will the population be in 50 years? Write the recursive and explicit formula. a. What are the first 3 terms? | b. Recursive: 9₁ = an c. Explicit: 9₁ = f(n) = d. What will the population be in 50 years? Remember, we have to find the 51st term to find the 50th year. f( )=
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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The United States population grows at a rate of 0.5% a year. We have around 328.2 million. What will the population be in 50 years? Write the recursive and explicit formula.
a. What are the first 3 terms?
b. Recursive:
a1 =
an=
c. Explicit
a1=
r=
f(n)=
d. What will the population be in 50 years? Remember, we have to find the 51* term to find the 50* year.
f( )=
![### Understanding Population Growth Through Sequences
The United States population grows at a rate of 0.5% a year. Currently, the population stands at approximately 328.2 million. This exercise explores how to calculate the population in 50 years using recursive and explicit formulas.
#### a. What are the first 3 terms?
To solve the sequence, identify the initial conditions and the pattern of growth.
#### b. Recursive Formula
Define the initial term and the recursive relationship:
- \( a_1 = \) Initial population
- \( a_n = \) Population in year \( n \)
The recursive formula will typically follow the pattern:
\[ a_n = a_{n-1} \cdot (1 + \text{growth rate}) \]
#### c. Explicit Formula
The explicit formula allows calculation of any term directly without recursion:
- \( a_1 = \) Initial population
- \( r = \text{growth rate} = 0.5\% \text{ or } 0.005 \)
The explicit formula is given by:
\[ f(n) = a_1 \cdot (1 + r)^{n-1} \]
#### d. Calculating Future Population
To find the population after 50 years, recognize that you need the 51st term in the sequence (representing year 50):
\[ f(51) = \text{Calculate using the explicit formula} \]
This exercise helps illustrate how sequences can model population growth over time.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3e0da6ca-d4b9-43ae-9bdc-db6e1c4ba965%2F0f2fcf98-95b5-4637-ba5f-ad7ecd991bb6%2Fey9v6z_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Population Growth Through Sequences
The United States population grows at a rate of 0.5% a year. Currently, the population stands at approximately 328.2 million. This exercise explores how to calculate the population in 50 years using recursive and explicit formulas.
#### a. What are the first 3 terms?
To solve the sequence, identify the initial conditions and the pattern of growth.
#### b. Recursive Formula
Define the initial term and the recursive relationship:
- \( a_1 = \) Initial population
- \( a_n = \) Population in year \( n \)
The recursive formula will typically follow the pattern:
\[ a_n = a_{n-1} \cdot (1 + \text{growth rate}) \]
#### c. Explicit Formula
The explicit formula allows calculation of any term directly without recursion:
- \( a_1 = \) Initial population
- \( r = \text{growth rate} = 0.5\% \text{ or } 0.005 \)
The explicit formula is given by:
\[ f(n) = a_1 \cdot (1 + r)^{n-1} \]
#### d. Calculating Future Population
To find the population after 50 years, recognize that you need the 51st term in the sequence (representing year 50):
\[ f(51) = \text{Calculate using the explicit formula} \]
This exercise helps illustrate how sequences can model population growth over time.
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