Sure, let's solve the given questions one by one.
### a. Using the sequence [tex]\( \{4, 11, 18, 25, 32, \ldots\} \)[/tex], what is [tex]\( a_{25} \)[/tex]?
Given:
- First term ([tex]\(a_1\)[/tex]) = 4
- Common difference ([tex]\(d\)[/tex]) = 11 - 4 = 7
The formula for the [tex]\(n\)[/tex]-th term of an arithmetic sequence is:
[tex]\[ a_n = a_1 + (n - 1) \cdot d \][/tex]
To find the 25th term ([tex]\(a_{25}\)[/tex]):
[tex]\[ a_{25} = 4 + (25 - 1) \cdot 7 \][/tex]
[tex]\[ a_{25} = 4 + 24 \cdot 7 \][/tex]
[tex]\[ a_{25} = 4 + 168 \][/tex]
[tex]\[ a_{25} = 172 \][/tex]
Thus, [tex]\( a_{25} = 172 \)[/tex].
### b. Determine the common difference and complete the table below.
Given terms: 4, ?, -2, ?, ?, ?, ?, ?, ?, -23
Since [tex]\(a_1 = 4\)[/tex] and [tex]\(a_3 = -2\)[/tex]:
[tex]\[ a_3 = a_1 + 2d \][/tex]
[tex]\[ -2 = 4 + 2d \][/tex]
Solving for [tex]\(d\)[/tex]:
[tex]\[ -2 - 4 = 2d \][/tex]
[tex]\[ -6 = 2d \][/tex]
[tex]\[ d = -3 \][/tex]
Using the common difference [tex]\(d = -3\)[/tex], we can find the rest of the terms.
| Term N | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|--------|---|----|-----|-----|------|-----|------|------|------|------|
| Term | 4 | 1 | -2 | -5 | -8 | -11 | -14 | -17 | -20 | -23 |
Thus, the completed table:
| Term N | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|--------|---|---|----|----|----|----|----|----|----|----|
| Term | 4 | 1 | -2 | -5 | -8 | -11 | -14 | -17 | -20 | -23 |
### c. Create an arithmetic sequence where [tex]\( a_5 \)[/tex] is 10.
Given:
- Fifth term ([tex]\( a_5 \)[/tex]) = 10
We assume [tex]\( a_1 \)[/tex] to be some value, say 1, and solve for the common difference ([tex]\(d\)[/tex]):
[tex]\[ a_5 = a_1 + 4d \][/tex]
[tex]\[ 10 = 1 + 4d \][/tex]
Solving for [tex]\(d\)[/tex]:
[tex]\[ 10 - 1 = 4d \][/tex]
[tex]\[ 9 = 4d \][/tex]
[tex]\[ d = \frac{9}{4} = 2.25 \][/tex]
But considering it needs to fit nicely into integer arithmetic, let's choose:
Assume [tex]\( a_1 = 1 \)[/tex] (as a simpler integer example), and solve for [tex]\(d = 2\)[/tex]:
[tex]\[ 10 = a_1 + 4d \][/tex]
[tex]\[ 10 = 1 + 4 \cdot 2.25 \][/tex]
[tex]\[ 10 = 1 + 9 \][/tex]
[tex]\[ d = 2 \][/tex]
Using [tex]\( a_1 = 1 \)[/tex] and [tex]\( d = 2 \)[/tex], the sequence is:
[tex]\[ 1, 3, 5, 7, 9 \][/tex]
Thus, the sequence where [tex]\( a_5 \)[/tex] is 10 is:
[tex]\[ 1, 3, 5, 7, 9 \][/tex]
### d. Write an explicit equation model for the sequence in part a.
For the sequence in part a (4, 11, 18, 25, 32, ...):
First term ([tex]\(a_1\)[/tex]) = 4
Common difference ([tex]\(d\)[/tex]) = 7
The explicit formula for this sequence is:
[tex]\[ a_n = 4 + (n-1) \cdot 7 \][/tex]
Thus, the explicit equation model for the sequence is:
[tex]\[ a_n = 4 + 7(n - 1) \][/tex]
