Question

3.7.11,15,19….. determine which option is the correct subscript notation for the explicit formula

Answer 1

To find the explicit formula for the sequence \(3, 7, 11, 15, 19, \ldots\), we first identify the pattern in the sequence.

1. **Identify the differences**:
\[
7 - 3 = 4, \quad 11 - 7 = 4, \quad 15 - 11 = 4, \quad 19 - 15 = 4
\]
The common difference is \(4\).

2. **Recognize the type of sequence**:
This is an arithmetic sequence where the first term \(a_1 = 3\) and the common difference \(d = 4\).

3. **Write the explicit formula**:
The general formula for the \(n\)-th term of an arithmetic sequence is given by:
\[
a_n = a_1 + (n - 1)d
\]
Substituting \(a_1 = 3\) and \(d = 4\) into the formula:
\[
a_n = 3 + (n - 1) \cdot 4
\]
Simplifying this:
\[
a_n = 3 + 4n - 4 = 4n - 1
\]

Thus, the explicit formula for the sequence is:
\[
a_n = 4n - 1
\]

If you are looking for the correct subscript notation to represent this formula, it would be:
\[
a_n = 4n - 1
\]
Where \(n\) is the index of the term in the sequence starting from \(1\).

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