Certainly! Let's determine the general term of the given sequence.
The given sequence is:
[tex]\[ 3, 5, 7, \ldots \][/tex]
To identify the general term, we need to determine two key components:
1. The first term ([tex]\(a_1\)[/tex])
2. The common difference ([tex]\(d\)[/tex])
Step 1: Identify the first term ([tex]\(a_1\)[/tex])
The first term of the sequence is quite explicitly the first number of the given sequence:
[tex]\[ a_1 = 3 \][/tex]
Step 2: Determine the common difference ([tex]\(d\)[/tex])
The common difference ([tex]\(d\)[/tex]) is found by subtracting the first term from the second term or any subsequent term from its previous term in the sequence:
[tex]\[ d = 5 - 3 = 2 \][/tex]
Step 3: Formulate the general term for the arithmetic sequence
For an arithmetic sequence, the nth term ([tex]\(a_n\)[/tex]) can be expressed using the following formula:
[tex]\[ a_n = a_1 + (n - 1) \cdot d \][/tex]
Plugging in the values of [tex]\(a_1\)[/tex] and [tex]\(d\)[/tex]:
[tex]\[ a_n = 3 + (n - 1) \cdot 2 \][/tex]
Let's simplify the expression:
[tex]\[ a_n = 3 + 2n - 2 \][/tex]
[tex]\[ a_n = 2n + 1 \][/tex]
Therefore, the general term for the given arithmetic sequence is:
[tex]\[ \boxed{a_n = 2n + 1} \][/tex]
