Answer :
To find the [tex]\(8^{\text{th}}\)[/tex] term of the given arithmetic sequence, we will follow these steps:
1. Identify the first term and the common difference:
- The first term of the sequence is given as [tex]\(f(1) = 4\)[/tex].
- The recursive formula provided is [tex]\(f(n) = f(n-1) + 7\)[/tex]. This tells us that the common difference [tex]\(d\)[/tex] is 7.
2. Use the formula for the [tex]\(n^{\text{th}}\)[/tex] term of an arithmetic sequence:
- The formula to find the [tex]\(n^{\text{th}}\)[/tex] term [tex]\(f(n)\)[/tex] of an arithmetic sequence is:
[tex]\[
f(n) = f(1) + (n-1) \times d
\][/tex]
where [tex]\(f(1)\)[/tex] is the first term, [tex]\(d\)[/tex] is the common difference, and [tex]\(n\)[/tex] is the term number.
3. Calculate the [tex]\(8^{\text{th}}\)[/tex] term:
- Substitute [tex]\(f(1) = 4\)[/tex], [tex]\(d = 7\)[/tex], and [tex]\(n = 8\)[/tex] into the formula:
[tex]\[
f(8) = 4 + (8-1) \times 7
\][/tex]
- Simplify the expression:
[tex]\[
f(8) = 4 + 7 \times 7
\][/tex]
[tex]\[
f(8) = 4 + 49
\][/tex]
[tex]\[
f(8) = 53
\][/tex]
Thus, the [tex]\(8^{\text{th}}\)[/tex] term of the sequence is [tex]\(\boxed{53}\)[/tex].
1. Identify the first term and the common difference:
- The first term of the sequence is given as [tex]\(f(1) = 4\)[/tex].
- The recursive formula provided is [tex]\(f(n) = f(n-1) + 7\)[/tex]. This tells us that the common difference [tex]\(d\)[/tex] is 7.
2. Use the formula for the [tex]\(n^{\text{th}}\)[/tex] term of an arithmetic sequence:
- The formula to find the [tex]\(n^{\text{th}}\)[/tex] term [tex]\(f(n)\)[/tex] of an arithmetic sequence is:
[tex]\[
f(n) = f(1) + (n-1) \times d
\][/tex]
where [tex]\(f(1)\)[/tex] is the first term, [tex]\(d\)[/tex] is the common difference, and [tex]\(n\)[/tex] is the term number.
3. Calculate the [tex]\(8^{\text{th}}\)[/tex] term:
- Substitute [tex]\(f(1) = 4\)[/tex], [tex]\(d = 7\)[/tex], and [tex]\(n = 8\)[/tex] into the formula:
[tex]\[
f(8) = 4 + (8-1) \times 7
\][/tex]
- Simplify the expression:
[tex]\[
f(8) = 4 + 7 \times 7
\][/tex]
[tex]\[
f(8) = 4 + 49
\][/tex]
[tex]\[
f(8) = 53
\][/tex]
Thus, the [tex]\(8^{\text{th}}\)[/tex] term of the sequence is [tex]\(\boxed{53}\)[/tex].