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]
