Let's analyse the arithmetic sequence given: 25, 31, 37, 43, 49...
To find the formula for an arithmetic sequence, we use the general form:
[tex]\[ f(n) = a + (n - 1) \cdot d \][/tex]
where [tex]\(a\)[/tex] is the first term of the sequence, [tex]\(d\)[/tex] is the common difference, and [tex]\(n\)[/tex] is the term number.
Step 1: Identify the first term and the common difference
From the given sequence:
- The first term [tex]\(a\)[/tex] is [tex]\(25\)[/tex].
- The common difference [tex]\(d\)[/tex] is calculated by subtracting the first term from the second term:
[tex]\[ d = 31 - 25 = 6 \][/tex]
Step 2: Write the general formula
Substituting [tex]\(a = 25\)[/tex] and [tex]\(d = 6\)[/tex] into the general formula, we get:
[tex]\[ f(n) = 25 + (n - 1) \cdot 6 \][/tex]
Step 3: Simplify the formula
Let's distribute and simplify the equation:
[tex]\[ f(n) = 25 + 6(n - 1) \][/tex]
[tex]\[ f(n) = 25 + 6n - 6 \][/tex]
[tex]\[ f(n) = 6n + 19 \][/tex]
Thus, the formula for the sequence based on our calculations simplifies to:
[tex]\[ f(n) = 6n + 19 \][/tex]
When we compare this formula to the options given:
1. [tex]\( f(n) = 25 + 6n \)[/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]
We observe that the closest match, after simplifying, to our formula [tex]\( f(n) = 6n + 19 \)[/tex] aligns with:
[tex]\[ f(n) = 19 + 6(n+1) \][/tex]
Therefore, the correct option is:
Option 4: [tex]\( f(n) = 19 + 6(n+1) \)[/tex]
