To determine the explicit form of the function represented by the sequence [tex]\( 3, 7, 11, 15 \)[/tex], first identify the type of sequence. This appears to be an arithmetic sequence since the difference between consecutive terms is constant.
### Step 1: Identify the Common Difference
Calculate the common difference ([tex]\(d\)[/tex]):
[tex]\[ 7 - 3 = 4 \][/tex]
[tex]\[ 11 - 7 = 4 \][/tex]
[tex]\[ 15 - 11 = 4 \][/tex]
The common difference is [tex]\(d = 4\)[/tex].
### Step 2: Identify the First Term
The first term of the sequence is [tex]\(a_1 = 3\)[/tex].
### Step 3: Write the General Formula for an Arithmetic Sequence
The general formula for an arithmetic sequence is:
[tex]\[ a_n = a_1 + (n-1)d \][/tex]
Plugging in the values we found:
[tex]\[ a_n = 3 + (n-1) \cdot 4 \][/tex]
[tex]\[ a_n = 3 + 4n - 4 \][/tex]
[tex]\[ a_n = 4n - 1 \][/tex]
### Step 4: Verify the Formula with Given Terms
Let's verify this formula by plugging in some values of [tex]\(n\)[/tex]:
- For [tex]\(n = 1\)[/tex]:
[tex]\[ a_1 = 4 \cdot 1 - 1 = 3 \][/tex]
- For [tex]\(n = 2\)[/tex]:
[tex]\[ a_2 = 4 \cdot 2 - 1 = 7 \][/tex]
- For [tex]\(n = 3\)[/tex]:
[tex]\[ a_3 = 4 \cdot 3 - 1 = 11 \][/tex]
- For [tex]\(n = 4\)[/tex]:
[tex]\[ a_4 = 4 \cdot 4 - 1 = 15 \][/tex]
The explicit formula [tex]\( a_n = 4n - 1 \)[/tex] fits all terms of the sequence perfectly.
### Conclusion: Identify the Correct Option
The correct form of the function is:
[tex]\[ B. \, f(n) = 4n - 1 \][/tex]
