Certainly! Let's analyze and solve the given sequence problem step-by-step.
We are given the sequence: [tex]\(2 + \sqrt{3}, 5 + \sqrt{3}, 8 + \sqrt{3}, \ldots\)[/tex]
### Step-by-Step Solution:
1. Identify the pattern:
- Let's denote the first term as [tex]\( a_1 \)[/tex].
- The first term is [tex]\( a_1 = 2 + \sqrt{3} \)[/tex].
- The second term is [tex]\( a_2 = 5 + \sqrt{3} \)[/tex].
- The third term is [tex]\( a_3 = 8 + \sqrt{3} \)[/tex].
2. Find the common difference [tex]\( d \)[/tex] of the arithmetic sequence:
- The difference between the second and the first term is:
[tex]\[
d = a_2 - a_1 = (5 + \sqrt{3}) - (2 + \sqrt{3}) = 3
\][/tex]
- The difference between the third and the second term is the same:
[tex]\[
d = a_3 - a_2 = (8 + \sqrt{3}) - (5 + \sqrt{3}) = 3
\][/tex]
Hence, we confirm that the common difference [tex]\( d = 3 \)[/tex].
3. General formula for the [tex]\( n^{\text{th}} \)[/tex] term of an arithmetic sequence:
- The [tex]\( n^{\text{th}} \)[/tex] term [tex]\( a_n \)[/tex] of an arithmetic sequence can be found using the formula:
[tex]\[
a_n = a_1 + (n - 1) \cdot d
\][/tex]
4. Substitute the known values into the formula:
- Here, [tex]\( a_1 = 2 + \sqrt{3} \)[/tex] and [tex]\( d = 3 \)[/tex].
- Substitute these values into the formula:
[tex]\[
a_n = 2 + \sqrt{3} + (n - 1) \cdot 3
\][/tex]
5. Simplify the expression:
- Distribute and simplify:
[tex]\[
a_n = 2 + \sqrt{3} + 3(n - 1)
\][/tex]
[tex]\[
a_n = 2 + \sqrt{3} + 3n - 3
\][/tex]
[tex]\[
a_n = 3n - 1 + \sqrt{3}
\][/tex]
Therefore, the [tex]\( n^{\text{th}} \)[/tex] term of the sequence is:
[tex]\[
a_n = 3n - 1 + \sqrt{3}
\][/tex]
This matches the given sequence and successfully gives us the solution.
