Welcome to Westonci.ca, the place where your questions are answered by a community of knowledgeable contributors. Ask your questions and receive accurate answers from professionals with extensive experience in various fields on our platform. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
Sagot :
To determine which option represents the formula for the given arithmetic sequence [tex]\(25, 31, 37, 43, 49, \ldots\)[/tex], we need to derive the general formula for the nth term of an arithmetic sequence.
An arithmetic sequence is defined by a first term ([tex]\(a\)[/tex]) and a common difference ([tex]\(d\)[/tex]). The nth term of an arithmetic sequence can be given by the formula:
[tex]\[ a_n = a + (n-1) \cdot d \][/tex]
where:
- [tex]\(a\)[/tex] is the first term of the sequence,
- [tex]\(d\)[/tex] is the common difference,
- [tex]\(n\)[/tex] is the term number.
Let's apply this to the given sequence:
1. Identify the first term ([tex]\(a\)[/tex]):
The first term of the sequence is [tex]\(25\)[/tex].
2. Determine the common difference ([tex]\(d\)[/tex]):
The common difference is the difference between any two consecutive terms. Let's calculate it using the first two terms:
[tex]\[ d = 31 - 25 = 6 \][/tex]
Now, we have:
- [tex]\(a = 25\)[/tex]
- [tex]\(d = 6\)[/tex]
Using the nth term formula for an arithmetic sequence:
[tex]\[ a_n = a + (n-1) \cdot d \][/tex]
Substitute [tex]\(a\)[/tex] and [tex]\(d\)[/tex] into the formula:
[tex]\[ a_n = 25 + (n-1) \cdot 6 \][/tex]
Simplify the expression:
[tex]\[ a_n = 25 + 6(n-1) \][/tex]
So the formula simplifies to:
[tex]\[ a_n = 25 + 6n - 6 \][/tex]
[tex]\[ a_n = 6n + 19 \][/tex]
Now, let's compare this with the given options to find the correct formula:
1. [tex]\(f(n) = 25 + 6(n)\)[/tex]
2. [tex]\(f(n) = 25 + 6(n+1)\)[/tex]
3. [tex]\(f(n) = 25 + 6(n-1)\)[/tex]
4. [tex]\(f(n) = 19 + 6(n+1)\)[/tex]
To match our derived formula:
[tex]\[ 6n + 19 \][/tex]
Let's analyze each option:
- Option 1: [tex]\(f(n) = 25 + 6(n)\)[/tex]:
This simplifies to [tex]\(25 + 6n\)[/tex], which does not match [tex]\(6n + 19\)[/tex].
- Option 2: [tex]\(f(n) = 25 + 6(n+1)\)[/tex]:
This simplifies to [tex]\(25 + 6n + 6\)[/tex], which simplifies further to [tex]\(6n + 31\)[/tex], which does not match [tex]\(6n + 19\)[/tex].
- Option 3: [tex]\(f(n) = 25 + 6(n-1)\)[/tex]:
This simplifies to [tex]\(25 + 6n - 6\)[/tex], which simplifies further to [tex]\(6n + 19\)[/tex], which matches our derived formula.
- Option 4: [tex]\(f(n) = 19 + 6(n+1)\)[/tex]:
This simplifies to [tex]\(19 + 6n + 6\)[/tex], which simplifies further to [tex]\(6n + 25\)[/tex], which does not match [tex]\(6n + 19\)[/tex].
Thus, the correct answer is the option which matches our derived formula:
[tex]\[ f(n) = 25 + 6(n-1) \][/tex]
So the correct answer is:
[tex]\[ \boxed{f(n) = 25 + 6(n-1)} \][/tex]
Therefore, the correct option is:
Option 3: [tex]\(f(n) = 25 + 6(n-1)\)[/tex]
An arithmetic sequence is defined by a first term ([tex]\(a\)[/tex]) and a common difference ([tex]\(d\)[/tex]). The nth term of an arithmetic sequence can be given by the formula:
[tex]\[ a_n = a + (n-1) \cdot d \][/tex]
where:
- [tex]\(a\)[/tex] is the first term of the sequence,
- [tex]\(d\)[/tex] is the common difference,
- [tex]\(n\)[/tex] is the term number.
Let's apply this to the given sequence:
1. Identify the first term ([tex]\(a\)[/tex]):
The first term of the sequence is [tex]\(25\)[/tex].
2. Determine the common difference ([tex]\(d\)[/tex]):
The common difference is the difference between any two consecutive terms. Let's calculate it using the first two terms:
[tex]\[ d = 31 - 25 = 6 \][/tex]
Now, we have:
- [tex]\(a = 25\)[/tex]
- [tex]\(d = 6\)[/tex]
Using the nth term formula for an arithmetic sequence:
[tex]\[ a_n = a + (n-1) \cdot d \][/tex]
Substitute [tex]\(a\)[/tex] and [tex]\(d\)[/tex] into the formula:
[tex]\[ a_n = 25 + (n-1) \cdot 6 \][/tex]
Simplify the expression:
[tex]\[ a_n = 25 + 6(n-1) \][/tex]
So the formula simplifies to:
[tex]\[ a_n = 25 + 6n - 6 \][/tex]
[tex]\[ a_n = 6n + 19 \][/tex]
Now, let's compare this with the given options to find the correct formula:
1. [tex]\(f(n) = 25 + 6(n)\)[/tex]
2. [tex]\(f(n) = 25 + 6(n+1)\)[/tex]
3. [tex]\(f(n) = 25 + 6(n-1)\)[/tex]
4. [tex]\(f(n) = 19 + 6(n+1)\)[/tex]
To match our derived formula:
[tex]\[ 6n + 19 \][/tex]
Let's analyze each option:
- Option 1: [tex]\(f(n) = 25 + 6(n)\)[/tex]:
This simplifies to [tex]\(25 + 6n\)[/tex], which does not match [tex]\(6n + 19\)[/tex].
- Option 2: [tex]\(f(n) = 25 + 6(n+1)\)[/tex]:
This simplifies to [tex]\(25 + 6n + 6\)[/tex], which simplifies further to [tex]\(6n + 31\)[/tex], which does not match [tex]\(6n + 19\)[/tex].
- Option 3: [tex]\(f(n) = 25 + 6(n-1)\)[/tex]:
This simplifies to [tex]\(25 + 6n - 6\)[/tex], which simplifies further to [tex]\(6n + 19\)[/tex], which matches our derived formula.
- Option 4: [tex]\(f(n) = 19 + 6(n+1)\)[/tex]:
This simplifies to [tex]\(19 + 6n + 6\)[/tex], which simplifies further to [tex]\(6n + 25\)[/tex], which does not match [tex]\(6n + 19\)[/tex].
Thus, the correct answer is the option which matches our derived formula:
[tex]\[ f(n) = 25 + 6(n-1) \][/tex]
So the correct answer is:
[tex]\[ \boxed{f(n) = 25 + 6(n-1)} \][/tex]
Therefore, the correct option is:
Option 3: [tex]\(f(n) = 25 + 6(n-1)\)[/tex]
We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.
