To determine the [tex]\( n^{\text{th}} \)[/tex] term of the given sequence [tex]\( 3, 5, 7, 9, 11, \ldots \)[/tex], let's analyze the sequence step-by-step and derive a general formula.
First, note that this sequence is an arithmetic sequence. In an arithmetic sequence, each term after the first is obtained by adding a constant difference (called the common difference) to the previous term.
### Step 1: Identify the First Term
The first term of the sequence is:
[tex]\[ a_1 = 3 \][/tex]
### Step 2: Determine the Common Difference
The common difference [tex]\( d \)[/tex] is the difference between consecutive terms:
[tex]\[ d = 5 - 3 = 2 \][/tex]
[tex]\[ d = 7 - 5 = 2 \][/tex]
[tex]\[ d = 9 - 7 = 2 \][/tex]
[tex]\[ d = 11 - 9 = 2 \][/tex]
So, the common difference [tex]\( d \)[/tex] is [tex]\( 2 \)[/tex].
### Step 3: Find the General Formula
The general formula for the [tex]\( n^{\text{th}} \)[/tex] term [tex]\( a_n \)[/tex] of an arithmetic sequence is:
[tex]\[ a_n = a_1 + (n - 1) \cdot d \][/tex]
### Step 4: Substitute the Known Values
Substitute [tex]\( a_1 = 3 \)[/tex] and [tex]\( d = 2 \)[/tex] into the general formula:
[tex]\[ a_n = 3 + (n - 1) \cdot 2 \][/tex]
### Step 5: Simplify the Formula
Let's simplify the expression:
[tex]\[ a_n = 3 + 2(n - 1) \][/tex]
[tex]\[ a_n = 3 + 2n - 2 \][/tex]
[tex]\[ a_n = 2n + 1 \][/tex]
So, the [tex]\( n^{\text{th}} \)[/tex] term of the sequence is given by the formula:
[tex]\[ \boxed{a_n = 2n + 1} \][/tex]
This formula will yield the [tex]\( n^{\text{th}} \)[/tex] term of the sequence for any positive integer [tex]\( n \)[/tex].
