A home cook needs to boil 1 liter of water on the stove. Water boils at [tex]100^{\circ} C[/tex].

\begin{tabular}{|l|c|c|c|}
\hline Time (minutes) & 3 & 4 & 5 \\
\hline Temperature [tex]\left(^{\circ} C \right)[/tex] & 38 & 46 & 84 \\
\hline
\end{tabular}

2. What is the common difference of the sequence represented in the table?
[tex]\square[/tex]

3. What is the explicit rule for the sequence?
[tex]\square[/tex]



Answer :

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.