Let's determine the explicit form of the function represented by the sequence [tex]\(3, 7, 11, 15\)[/tex].
Firstly, observe the sequence and consider the differences between successive terms:
- [tex]\(7 - 3 = 4\)[/tex]
- [tex]\(11 - 7 = 4\)[/tex]
- [tex]\(15 - 11 = 4\)[/tex]
The difference between consecutive terms is constant and equal to 4. This indicates that the sequence is arithmetic with a common difference [tex]\(d = 4\)[/tex].
In an arithmetic sequence, the [tex]\(n\)[/tex]-th term [tex]\(f(n)\)[/tex] can be represented using the formula:
[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
- [tex]\(n\)[/tex] is the term number
For the given sequence:
- The first term [tex]\(a = 3\)[/tex]
- The common difference [tex]\(d = 4\)[/tex]
Substitute these values into the formula to express [tex]\(f(n)\)[/tex]:
[tex]\[ f(n) = 3 + (n-1) \cdot 4 \][/tex]
[tex]\[ f(n) = 3 + 4n - 4 \][/tex]
[tex]\[ f(n) = 4n - 1 \][/tex]
So the explicit form of the function is [tex]\( f(n) = 4n - 1 \)[/tex].
Next, we should verify this explicit form by calculating the first few terms:
1. For [tex]\( n = 1 \)[/tex]:
[tex]\[ f(1) = 4 \cdot 1 - 1 = 4 - 1 = 3 \][/tex]
2. For [tex]\( n = 2 \)[/tex]:
[tex]\[ f(2) = 4 \cdot 2 - 1 = 8 - 1 = 7 \][/tex]
3. For [tex]\( n = 3 \)[/tex]:
[tex]\[ f(3) = 4 \cdot 3 - 1 = 12 - 1 = 11 \][/tex]
4. For [tex]\( n = 4 \)[/tex]:
[tex]\[ f(4) = 4 \cdot 4 - 1 = 16 - 1 = 15 \][/tex]
These calculated values correspond to the given sequence [tex]\(3, 7, 11, 15\)[/tex], confirming our formula.
Therefore, the correct answer is:
[tex]\[ \boxed{f(n) = 4n - 1} \][/tex]
