Answer :
### Solution:
#### Part (a): Find the 25th term [tex]\( a_{25} \)[/tex] of the sequence [tex]\( \{4, 11, 18, 25, 32, \ldots\} \)[/tex]
The sequence given is an arithmetic sequence with the first term [tex]\( a_1 = 4 \)[/tex] and a common difference [tex]\( d = 7 \)[/tex] (since [tex]\( 11 - 4 = 7 \)[/tex]).
The general formula for the [tex]\( n \)[/tex]-th term [tex]\( a_n \)[/tex] of an arithmetic sequence is:
[tex]\[ a_n = a_1 + (n - 1) \cdot d \][/tex]
Substituting [tex]\( a_1 = 4 \)[/tex], [tex]\( d = 7 \)[/tex], and [tex]\( n = 25 \)[/tex] into the formula, we get:
[tex]\[ a_{25} = 4 + (25 - 1) \cdot 7 \][/tex]
Simplifying:
[tex]\[ a_{25} = 4 + 24 \cdot 7 \][/tex]
[tex]\[ a_{25} = 4 + 168 \][/tex]
[tex]\[ a_{25} = 172 \][/tex]
So, the 25th term [tex]\( a_{25} \)[/tex] is [tex]\( \boxed{172} \)[/tex].
#### Part (b): Determine the common difference and complete the table below
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline \text{Term } \# & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \text{Term } a & 4 & 1 & -2 & -5 & -8 & -11 & -14 & -17 & -20 & -23 \\ \hline \end{array} \][/tex]
To determine the common difference [tex]\( d \)[/tex]:
Notice the pattern from the sequence provided in the table:
- [tex]\( 1 - 4 = -3 \)[/tex]
- [tex]\( -2 - 1 = -3 \)[/tex]
- [tex]\( -5 - (-2) = -3 \)[/tex]
- [tex]\( -8 - (-5) = -3 \)[/tex]
- [tex]\( -11 - (-8) = -3 \)[/tex]
So, the common difference [tex]\( d = -3 \)[/tex].
#### Part (c): Create an arithmetic sequence where [tex]\( a_s \)[/tex] is 10
Given that [tex]\( a_s = 10 \)[/tex], we want to construct an arithmetic sequence such that one of its terms has this value. Let's use the sequence given in part (a) where the first term [tex]\( a_1 = 4 \)[/tex] and the common difference [tex]\( d = 7 \)[/tex].
To find the position [tex]\( s \)[/tex] where the term [tex]\( a_s \)[/tex] is 10, we use the general term formula:
[tex]\[ a_s = a_1 + (s - 1) \cdot d \][/tex]
Substituting [tex]\( a_s = 10 \)[/tex], [tex]\( a_1 = 4 \)[/tex], and [tex]\( d = 7 \)[/tex]:
[tex]\[ 10 = 4 + (s - 1) \cdot 7 \][/tex]
Solving for [tex]\( s \)[/tex]:
[tex]\[ 10 = 4 + 7(s - 1) \][/tex]
[tex]\[ 10 - 4 = 7(s - 1) \][/tex]
[tex]\[ 6 = 7(s - 1) \][/tex]
[tex]\[ \frac{6}{7} = s - 1 \][/tex]
[tex]\[ s = \frac{6}{7} + 1 \][/tex]
[tex]\[ s = \frac{13}{7} \][/tex]
So in this sequence, the term value 10 is found at the position [tex]\( s = \frac{13}{7} \)[/tex], approximately [tex]\( s = 1.857 \)[/tex].
#### Part (d): Write an explicit equation model for the sequence in part (a)
For the arithmetic sequence given in part (a) [tex]\( \{4, 11, 18, 25, 32, \ldots\} \)[/tex], with the first term [tex]\( a_1 = 4 \)[/tex] and the common difference [tex]\( d = 7 \)[/tex], the explicit equation for the [tex]\( n \)[/tex]-th term [tex]\( a_n \)[/tex] is:
[tex]\[ a_n = 4 + (n - 1) \cdot 7 \][/tex]
Therefore, the explicit equation model for the sequence is:
[tex]\[ \boxed{a_n = 4 + (n - 1) \cdot 7} \][/tex]
This completes the solution to all the parts of the given question.
#### Part (a): Find the 25th term [tex]\( a_{25} \)[/tex] of the sequence [tex]\( \{4, 11, 18, 25, 32, \ldots\} \)[/tex]
The sequence given is an arithmetic sequence with the first term [tex]\( a_1 = 4 \)[/tex] and a common difference [tex]\( d = 7 \)[/tex] (since [tex]\( 11 - 4 = 7 \)[/tex]).
The general formula for the [tex]\( n \)[/tex]-th term [tex]\( a_n \)[/tex] of an arithmetic sequence is:
[tex]\[ a_n = a_1 + (n - 1) \cdot d \][/tex]
Substituting [tex]\( a_1 = 4 \)[/tex], [tex]\( d = 7 \)[/tex], and [tex]\( n = 25 \)[/tex] into the formula, we get:
[tex]\[ a_{25} = 4 + (25 - 1) \cdot 7 \][/tex]
Simplifying:
[tex]\[ a_{25} = 4 + 24 \cdot 7 \][/tex]
[tex]\[ a_{25} = 4 + 168 \][/tex]
[tex]\[ a_{25} = 172 \][/tex]
So, the 25th term [tex]\( a_{25} \)[/tex] is [tex]\( \boxed{172} \)[/tex].
#### Part (b): Determine the common difference and complete the table below
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline \text{Term } \# & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \text{Term } a & 4 & 1 & -2 & -5 & -8 & -11 & -14 & -17 & -20 & -23 \\ \hline \end{array} \][/tex]
To determine the common difference [tex]\( d \)[/tex]:
Notice the pattern from the sequence provided in the table:
- [tex]\( 1 - 4 = -3 \)[/tex]
- [tex]\( -2 - 1 = -3 \)[/tex]
- [tex]\( -5 - (-2) = -3 \)[/tex]
- [tex]\( -8 - (-5) = -3 \)[/tex]
- [tex]\( -11 - (-8) = -3 \)[/tex]
So, the common difference [tex]\( d = -3 \)[/tex].
#### Part (c): Create an arithmetic sequence where [tex]\( a_s \)[/tex] is 10
Given that [tex]\( a_s = 10 \)[/tex], we want to construct an arithmetic sequence such that one of its terms has this value. Let's use the sequence given in part (a) where the first term [tex]\( a_1 = 4 \)[/tex] and the common difference [tex]\( d = 7 \)[/tex].
To find the position [tex]\( s \)[/tex] where the term [tex]\( a_s \)[/tex] is 10, we use the general term formula:
[tex]\[ a_s = a_1 + (s - 1) \cdot d \][/tex]
Substituting [tex]\( a_s = 10 \)[/tex], [tex]\( a_1 = 4 \)[/tex], and [tex]\( d = 7 \)[/tex]:
[tex]\[ 10 = 4 + (s - 1) \cdot 7 \][/tex]
Solving for [tex]\( s \)[/tex]:
[tex]\[ 10 = 4 + 7(s - 1) \][/tex]
[tex]\[ 10 - 4 = 7(s - 1) \][/tex]
[tex]\[ 6 = 7(s - 1) \][/tex]
[tex]\[ \frac{6}{7} = s - 1 \][/tex]
[tex]\[ s = \frac{6}{7} + 1 \][/tex]
[tex]\[ s = \frac{13}{7} \][/tex]
So in this sequence, the term value 10 is found at the position [tex]\( s = \frac{13}{7} \)[/tex], approximately [tex]\( s = 1.857 \)[/tex].
#### Part (d): Write an explicit equation model for the sequence in part (a)
For the arithmetic sequence given in part (a) [tex]\( \{4, 11, 18, 25, 32, \ldots\} \)[/tex], with the first term [tex]\( a_1 = 4 \)[/tex] and the common difference [tex]\( d = 7 \)[/tex], the explicit equation for the [tex]\( n \)[/tex]-th term [tex]\( a_n \)[/tex] is:
[tex]\[ a_n = 4 + (n - 1) \cdot 7 \][/tex]
Therefore, the explicit equation model for the sequence is:
[tex]\[ \boxed{a_n = 4 + (n - 1) \cdot 7} \][/tex]
This completes the solution to all the parts of the given question.
