To determine the nth term of the given sequence 7, 9, 11, 13, 15, ..., we first need to analyze its structure.
1. Identify the type of sequence: The given sequence is an arithmetic sequence because the difference between consecutive terms is constant. To confirm, let's calculate the common difference:
[tex]\(9 - 7 = 2\)[/tex]
[tex]\(11 - 9 = 2\)[/tex]
[tex]\(13 - 11 = 2\)[/tex]
[tex]\(15 - 13 = 2\)[/tex]
Hence, the common difference [tex]\(d = 2\)[/tex].
2. Formulate the nth term:
The general formula for the nth term of an arithmetic sequence is:
[tex]\[ a_n = a_1 + (n - 1) \cdot d \][/tex]
Here, [tex]\(a_1\)[/tex] is the first term of the sequence, which is 7, and [tex]\(d\)[/tex] is the common difference, which is 2.
3. Substitute the values:
[tex]\[ a_n = 7 + (n - 1) \cdot 2 \][/tex]
4. Simplify the expression:
[tex]\[ a_n = 7 + 2(n - 1) \][/tex]
[tex]\[ a_n = 7 + 2n - 2 \][/tex]
[tex]\[ a_n = 2n + 5 \][/tex]
5. Determine which option matches:
From the simplified formula, we have [tex]\( a_n = 2n + 5 \)[/tex].
Therefore, among the given options, the correct nth term of the sequence is:
[tex]\[ a_n = 2n + 5 \][/tex]
This corresponds to the second option. Hence, the answer is the second option:
[tex]\[ \boxed{2} \][/tex]
