Sure, let's solve this step-by-step.
Given:
- The fourth term of the arithmetic sequence ([tex]\(a_4\)[/tex]) is 9.
- The common difference ([tex]\(d\)[/tex]) is 2.
We need to find the third term ([tex]\(a_3\)[/tex]) and the fifth term ([tex]\(a_5\)[/tex]).
Step 1: Finding the Third Term ([tex]\(a_3\)[/tex])
To find the third term of the sequence, we use the definition of an arithmetic sequence, where each term differs from the previous one by the common difference [tex]\(d\)[/tex].
The formula for the [tex]\(n\)[/tex]th term of an arithmetic sequence is given by:
[tex]\[
a_n = a_{n-1} + d
\][/tex]
We can rearrange this formula to find the previous term:
[tex]\[
a_{n-1} = a_n - d
\][/tex]
Substituting [tex]\(n = 4\)[/tex] for the fourth term:
[tex]\[
a_3 = a_4 - d
\][/tex]
Given [tex]\(a_4 = 9\)[/tex] and [tex]\(d = 2\)[/tex]:
[tex]\[
a_3 = 9 - 2
\][/tex]
[tex]\[
a_3 = 7
\][/tex]
Step 2: Finding the Fifth Term ([tex]\(a_5\)[/tex])
Similarly, to find the fifth term of the sequence, we use the formula for the [tex]\(n\)[/tex]th term again:
[tex]\[
a_{n+1} = a_n + d
\][/tex]
Substituting [tex]\(n = 4\)[/tex] for the fourth term:
[tex]\[
a_5 = a_4 + d
\][/tex]
Given [tex]\(a_4 = 9\)[/tex] and [tex]\(d = 2\)[/tex]:
[tex]\[
a_5 = 9 + 2
\][/tex]
[tex]\[
a_5 = 11
\][/tex]
Conclusion:
The third term of the arithmetic sequence is [tex]\(7\)[/tex] and the fifth term is [tex]\(11\)[/tex].
