To address the questions, let's first analyze the sequence given: [tex]\(3, 5, 7, \ldots\)[/tex].
### Part a. Determine the general term of the arithmetic sequence.
Let's start by recognizing that this is an arithmetic sequence. An arithmetic sequence is one in which each term after the first is obtained by adding a constant difference to the preceding term. Here's how you can figure out the general term:
1. Identify the first term ([tex]\(a_1\)[/tex]):
The first term [tex]\( a_1 = 3 \)[/tex].
2. Determine the common difference ([tex]\(d\)[/tex]):
To find the common difference, subtract the first term from the second term:
[tex]\[
d = 5 - 3 = 2
\][/tex]
3. Formulate the general term ([tex]\(a_n\)[/tex]):
The formula for the [tex]\(n\)[/tex]-th term of an arithmetic sequence is:
[tex]\[
a_n = a_1 + (n - 1) \cdot d
\][/tex]
Plugging in the values we found for [tex]\(a_1\)[/tex] and [tex]\(d\)[/tex]:
[tex]\[
a_n = 3 + (n - 1) \cdot 2
\][/tex]
Thus, the general term of the arithmetic sequence is:
[tex]\[
a_n = 3 + (n - 1) \cdot 2
\][/tex]
### Part b. Determine which term is equal to 71.
To find which term in the sequence is equal to 71, we need to set the general term equal to 71 and solve for [tex]\(n\)[/tex]:
1. Set the general term equal to 71:
[tex]\[
71 = 3 + (n - 1) \cdot 2
\][/tex]
2. Solve for [tex]\(n\)[/tex]:
[tex]\[
71 = 3 + 2(n - 1)
\][/tex]
First, subtract 3 from both sides of the equation:
[tex]\[
71 - 3 = 2(n - 1)
\][/tex]
[tex]\[
68 = 2(n - 1)
\][/tex]
Next, divide both sides by 2:
[tex]\[
\frac{68}{2} = n - 1
\][/tex]
[tex]\[
34 = n - 1
\][/tex]
Finally, add 1 to both sides to solve for [tex]\(n\)[/tex]:
[tex]\[
34 + 1 = n
\][/tex]
[tex]\[
n = 35
\][/tex]
Therefore, the 35th term in the sequence is equal to 71.
