Certainly! Let's solve the problem step-by-step.
### Step 1: Understanding the Problem
We have a table showing the temperature of water at different times:
[tex]\[
\begin{array}{|l|c|c|c|}
\hline
\text{Time (minutes)} & 3 & 4 & 5 \\
\hline
\text{Temperature} \left(^{\circ} \mathrm{C}\right) & 38 & 46 & 84 \\
\hline
\end{array}
\][/tex]
We need to determine:
1. The common difference in the sequence of temperatures.
2. The explicit rule for the sequence.
### Step 2: Calculate the Common Difference of the Sequence
The common difference in an arithmetic sequence is the difference between successive terms. We will calculate the differences between consecutive temperatures:
1. The difference between the temperature at 4 minutes and the temperature at 3 minutes:
[tex]\[
46 - 38 = 8
\][/tex]
2. The difference between the temperature at 5 minutes and the temperature at 4 minutes:
[tex]\[
84 - 46 = 38
\][/tex]
The calculated differences are:
- [tex]\(8\)[/tex] (from [tex]\(T(4) - T(3)\)[/tex])
- [tex]\(38\)[/tex] (from [tex]\(T(5) - T(4)\)[/tex])
Since the differences are not the same, there is no common difference in this sequence. The temperatures do not form a strict arithmetic sequence.
To answer the specific questions:
1. Common Difference of the Sequence: There is no common difference in this sequence.
### Step 3: Determine the Explicit Rule for the Sequence
An explicit rule for an arithmetic sequence takes the form:
[tex]\[
T(n) = a + (n-1) \cdot d
\][/tex]
where:
- [tex]\(a\)[/tex] is the first term.
- [tex]\(d\)[/tex] is the common difference.
- [tex]\(n\)[/tex] is the position in the sequence.
Since there is no common difference [tex]\(d\)[/tex], we cannot form an explicit arithmetic rule. Therefore, there is no explicit rule that matches the given data as an arithmetic sequence.
To answer the second question:
2. Explicit Rule for the Sequence: There is no explicit rule because the sequence is not arithmetic.
