Sure, let's find the recursive formulas for both given arithmetic sequences step-by-step.
### Problem 27
We are given the following terms in an arithmetic sequence:
- [tex]\(a_{18} = 3362\)[/tex]
- [tex]\(a_{38} = 7362\)[/tex]
To define the recursive formula for this arithmetic sequence, we need to find the common difference [tex]\(d\)[/tex] and the first term [tex]\(a_1\)[/tex].
1. Calculate the common difference [tex]\(d\)[/tex]:
The common difference [tex]\(d\)[/tex] in an arithmetic sequence can be found using the formula:
[tex]\[
d = \frac{a_m - a_n}{m - n}
\][/tex]
Here, [tex]\(m = 38\)[/tex], [tex]\(n = 18\)[/tex], [tex]\(a_{38} = 7362\)[/tex], and [tex]\(a_{18} = 3362\)[/tex]:
[tex]\[
d = \frac{7362 - 3362}{38 - 18} = \frac{4000}{20} = 200
\][/tex]
2. Calculate the first term [tex]\(a_1\)[/tex]:
We know the general form of the [tex]\(n'th\)[/tex] term of an arithmetic sequence:
[tex]\[
a_n = a_1 + (n - 1)d
\][/tex]
Using [tex]\(a_{18} = 3362\)[/tex]:
[tex]\[
3362 = a_1 + (18 - 1) \cdot 200
\][/tex]
Simplify and solve for [tex]\(a_1\)[/tex]:
[tex]\[
3362 = a_1 + 17 \cdot 200
\][/tex]
[tex]\[
3362 = a_1 + 3400
\][/tex]
[tex]\[
a_1 = 3362 - 3400 = -38
\][/tex]
3. Define the recursive formula:
The recursive formula for an arithmetic sequence is given by:
[tex]\[
a_n = a_{n-1} + d
\][/tex]
In this case, [tex]\(d = 200\)[/tex]:
[tex]\[
a_n = a_{n-1} + 200
\][/tex]
### Problem 28
We are given the following terms in an arithmetic sequence:
- [tex]\(a_{18} = 44.3\)[/tex]
- [tex]\(a_{33} = 84.8\)[/tex]
1. Calculate the common difference [tex]\(d\)[/tex]:
The common difference [tex]\(d\)[/tex] in this case can be found using the same formula:
[tex]\[
d = \frac{a_{33} - a_{18}}{33 - 18} = \frac{84.8 - 44.3}{33 - 18} = \frac{40.5}{15} = 2.7
\][/tex]
2. Calculate the first term [tex]\(a_1\)[/tex]:
Using [tex]\(a_{18} = 44.3\)[/tex]:
[tex]\[
44.3 = a_1 + (18 - 1) \cdot 2.7
\][/tex]
Simplify and solve for [tex]\(a_1\)[/tex]:
[tex]\[
44.3 = a_1 + 17 \cdot 2.7
\][/tex]
[tex]\[
44.3 = a_1 + 45.9
\][/tex]
[tex]\[
a_1 = 44.3 - 45.9 = -1.6
\][/tex]
3. Define the recursive formula:
Similarly, the recursive formula for this arithmetic sequence is:
[tex]\[
a_n = a_{n-1} + d
\][/tex]
In this case, [tex]\(d = 2.7\)[/tex]:
[tex]\[
a_n = a_{n-1} + 2.7
\][/tex]
To summarize:
For Problem 27:
- Common difference [tex]\(d = 200\)[/tex]
- First term [tex]\(a_1 = -38\)[/tex]
- Recursive formula: [tex]\(\boxed{a_n = a_{n-1} + 200}\)[/tex]
For Problem 28:
- Common difference [tex]\(d = 2.7\)[/tex]
- First term [tex]\(a_1 = -1.6\)[/tex]
- Recursive formula: [tex]\(\boxed{a_n = a_{n-1} + 2.7}\)[/tex]
