An Arithmetic Sequence Begins With ${ 25, 31, 37, 43, 49, \ldots\$} Which Option Below Represents The Formula For The Sequence?A. { F(n) = 25 + 6n$}$B. { F(n) = 25 + 6(n+1)$}$C. { F(n) = 25 + 6(n-1)$}$D.
An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. In other words, if we have a sequence of numbers, and the difference between each pair of consecutive numbers is the same, then it is an arithmetic sequence. The given sequence $<span class="katex-error" title="ParseError' at position 29: …3, 49, \ldots$}̲" style="color:#cc0000">25, 31, 37, 43, 49, \ldots$} is an example of an arithmetic sequence.
Understanding the Formula
The formula for an arithmetic sequence is given by {f(n) = a + (n-1)d$}$, where {a$}$ is the first term of the sequence, {d$}$ is the common difference, and {n$}$ is the term number. In the given sequence, the first term is 25, and the common difference is 6.
Analyzing the Options
Let's analyze the options given to determine which one represents the formula for the sequence.
Option A: {f(n) = 25 + 6n$}$
This option suggests that the formula for the sequence is {f(n) = 25 + 6n$}$. However, this formula does not take into account the fact that the common difference is 6, and the first term is 25. The correct formula should be {f(n) = 25 + 6(n-1)$}$, which is option C.
Option B: {f(n) = 25 + 6(n+1)$}$
This option suggests that the formula for the sequence is {f(n) = 25 + 6(n+1)$}$. However, this formula is incorrect because it adds 1 to the term number, which is not necessary.
Option C: {f(n) = 25 + 6(n-1)$}$
This option suggests that the formula for the sequence is {f(n) = 25 + 6(n-1)$}$. This formula is correct because it takes into account the fact that the common difference is 6, and the first term is 25.
Option D: {f(n) = 25 + 6(n-1) + 1$}$
This option suggests that the formula for the sequence is {f(n) = 25 + 6(n-1) + 1$}$. However, this formula is incorrect because it adds 1 to the formula, which is not necessary.
Conclusion
In conclusion, the correct formula for the given arithmetic sequence is {f(n) = 25 + 6(n-1)$}$. This formula takes into account the fact that the common difference is 6, and the first term is 25. The other options are incorrect because they do not accurately represent the formula for the sequence.
Understanding the Formula: A Step-by-Step Guide
To understand the formula for an arithmetic sequence, let's break it down step by step.
Step 1: Identify the First Term
The first term of the sequence is the first number in the sequence. In the given sequence, the first term is 25.
Step 2: Identify the Common Difference
The common difference is the difference between any two consecutive terms in the sequence. In the given sequence, the common difference is 6.
Step 3: Write the Formula
The formula for an arithmetic sequence is given by {f(n) = a + (n-1)d$}$, where {a$}$ is the first term, {d$}$ is the common difference, and {n$}$ is the term number.
Step 4: Plug in the Values
To write the formula for the given sequence, we need to plug in the values of the first term and the common difference. The first term is 25, and the common difference is 6.
Step 5: Simplify the Formula
Once we have plugged in the values, we can simplify the formula to get the final result.
Example: Finding the 5th Term
To find the 5th term of the sequence, we can use the formula {f(n) = 25 + 6(n-1)$}$. We need to plug in the value of n, which is 5.
Step 1: Plug in the Value of n
{f(5) = 25 + 6(5-1)$}$
Step 2: Simplify the Formula
{f(5) = 25 + 6(4)$}$
Step 3: Multiply 6 and 4
{f(5) = 25 + 24$}$
Step 4: Add 25 and 24
{f(5) = 49$}$
Therefore, the 5th term of the sequence is 49.
Real-World Applications
Arithmetic sequences have many real-world applications. Here are a few examples:
Example 1: Population Growth
The population of a city can be modeled using an arithmetic sequence. If the population of the city is increasing by 100 people per year, and the current population is 100,000, then the population after 5 years can be found using the formula {f(n) = 100,000 + 100(n-1)$}$.
Example 2: Sales Revenue
The sales revenue of a company can be modeled using an arithmetic sequence. If the sales revenue is increasing by 10% per year, and the current sales revenue is $100,000, then the sales revenue after 5 years can be found using the formula {f(n) = 100,000 + 0.1(100,000)(n-1)$}$.
Example 3: Temperature
The temperature of a city can be modeled using an arithmetic sequence. If the temperature is increasing by 2°C per hour, and the current temperature is 20°C, then the temperature after 5 hours can be found using the formula {f(n) = 20 + 2(n-1)$}$.
Conclusion
Q: What is an arithmetic sequence?
A: An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. In other words, if we have a sequence of numbers, and the difference between each pair of consecutive numbers is the same, then it is an arithmetic sequence.
Q: What is the formula for an arithmetic sequence?
A: The formula for an arithmetic sequence is given by {f(n) = a + (n-1)d$}$, where {a$}$ is the first term, {d$}$ is the common difference, and {n$}$ is the term number.
Q: How do I find the nth term of an arithmetic sequence?
A: To find the nth term of an arithmetic sequence, you can use the formula {f(n) = a + (n-1)d$}$. Simply plug in the values of the first term, the common difference, and the term number.
Q: How do I find the common difference of an arithmetic sequence?
A: To find the common difference of an arithmetic sequence, you can subtract any two consecutive terms. For example, if the sequence is ${2, 5, 8, 11, \ldots\$}, then the common difference is ${5 - 2 = 3\$}.
Q: How do I find the first term of an arithmetic sequence?
A: To find the first term of an arithmetic sequence, you can use the formula {f(n) = a + (n-1)d$}$. Simply plug in the values of the nth term, the common difference, and the term number, and solve for the first term.
Q: What is the relationship between the first term, the common difference, and the nth term of an arithmetic sequence?
A: The relationship between the first term, the common difference, and the nth term of an arithmetic sequence is given by the formula {f(n) = a + (n-1)d$}$. This formula shows that the nth term is equal to the first term plus the product of the common difference and the term number minus one.
Q: Can an arithmetic sequence have a negative common difference?
A: Yes, an arithmetic sequence can have a negative common difference. For example, the sequence {-2, -5, -8, -11, \ldots$}$ has a negative common difference of {-3$}$.
Q: Can an arithmetic sequence have a zero common difference?
A: Yes, an arithmetic sequence can have a zero common difference. For example, the sequence ${1, 1, 1, 1, \ldots\$} has a zero common difference.
Q: What is the significance of the term number in an arithmetic sequence?
A: The term number in an arithmetic sequence represents the position of the term in the sequence. For example, the first term is the 1st term, the second term is the 2nd term, and so on.
Q: Can an arithmetic sequence have a fractional term number?
A: No, an arithmetic sequence cannot have a fractional term number. The term number must be a whole number.
Q: Can an arithmetic sequence have a negative term number?
A: No, an arithmetic sequence cannot have a negative term number. The term number must be a whole number greater than or equal to 1.
Q: What is the relationship between the sum of an arithmetic sequence and the first term, the common difference, and the number of terms?
A: The relationship between the sum of an arithmetic sequence and the first term, the common difference, and the number of terms is given by the formula {S_n = \frac{n}{2}(2a + (n-1)d)$}$. This formula shows that the sum of the sequence is equal to the product of the number of terms, the first term, and the common difference, divided by 2.
Q: Can an arithmetic sequence have a sum that is not a whole number?
A: Yes, an arithmetic sequence can have a sum that is not a whole number. For example, the sequence ${1, 2, 3, 4, \ldots\$} has a sum of {\frac{5}{2}(2 + 4) = 15$}$, which is not a whole number.
Q: Can an arithmetic sequence have a sum that is negative?
A: Yes, an arithmetic sequence can have a sum that is negative. For example, the sequence {-1, -2, -3, -4, \ldots$}$ has a sum of {\frac{5}{2}(-2 + 4) = -5$}$, which is negative.
Q: What is the significance of the sum of an arithmetic sequence?
A: The sum of an arithmetic sequence represents the total value of the sequence. For example, if the sequence represents the sales revenue of a company, then the sum of the sequence represents the total sales revenue.
Q: Can an arithmetic sequence have a sum that is zero?
A: Yes, an arithmetic sequence can have a sum that is zero. For example, the sequence {-1, 1, -1, 1, \ldots$}$ has a sum of {\frac{5}{2}(-2 + 2) = 0$}$, which is zero.
Conclusion
In conclusion, arithmetic sequences are a fundamental concept in mathematics that have many real-world applications. The formula for an arithmetic sequence is given by {f(n) = a + (n-1)d$}$, where {a$}$ is the first term, {d$}$ is the common difference, and {n$}$ is the term number. By understanding the formula and how to use it, we can solve many problems in mathematics and real-world applications.