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Answer :
To solve this problem, let's examine the pattern in the data from the table:
```
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 5 \\
2 & 25 \\
3 & 125 \\
4 & 625 \\
5 & 3125 \\
\hline
\end{array}
\][/tex]
```
Notice that as [tex]\( x \)[/tex] increases, each corresponding [tex]\( y \)[/tex] value in the table is a result of a certain pattern. Let's identify this pattern step-by-step:
1. Identify the pattern:
- Look at the values of [tex]\( y \)[/tex] for each [tex]\( x \)[/tex].
- When [tex]\( x = 1 \)[/tex], [tex]\( y = 5 \)[/tex].
- When [tex]\( x = 2 \)[/tex], [tex]\( y = 25 \)[/tex].
- When [tex]\( x = 3 \)[/tex], [tex]\( y = 125 \)[/tex].
- When [tex]\( x = 4 \)[/tex], [tex]\( y = 625 \)[/tex].
- When [tex]\( x = 5 \)[/tex], [tex]\( y = 3125 \)[/tex].
2. Determine the relationship:
- Note that the value of [tex]\( y \)[/tex] seems to be a power of 5. Specifically, each [tex]\( y \)[/tex] value can be expressed as 5 raised to the power of the corresponding [tex]\( x \)[/tex] value.
- For each [tex]\( x \)[/tex], [tex]\( y = 5^x \)[/tex].
3. Verify the pattern:
- Calculate [tex]\( y \)[/tex] using [tex]\( y = 5^x \)[/tex] for each given [tex]\( x \)[/tex]:
- For [tex]\( x = 1 \)[/tex], [tex]\( y = 5^1 = 5 \)[/tex].
- For [tex]\( x = 2 \)[/tex], [tex]\( y = 5^2 = 25 \)[/tex].
- For [tex]\( x = 3 \)[/tex], [tex]\( y = 5^3 = 125 \)[/tex].
- For [tex]\( x = 4 \)[/tex], [tex]\( y = 5^4 = 625 \)[/tex].
- For [tex]\( x = 5 \)[/tex], [tex]\( y = 5^5 = 3125 \)[/tex].
The calculated [tex]\( y \)[/tex] values match those in the table perfectly, following the formula [tex]\( y = 5^x \)[/tex]. Therefore, the solution confirms the pattern and relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the given table.
```
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 5 \\
2 & 25 \\
3 & 125 \\
4 & 625 \\
5 & 3125 \\
\hline
\end{array}
\][/tex]
```
Notice that as [tex]\( x \)[/tex] increases, each corresponding [tex]\( y \)[/tex] value in the table is a result of a certain pattern. Let's identify this pattern step-by-step:
1. Identify the pattern:
- Look at the values of [tex]\( y \)[/tex] for each [tex]\( x \)[/tex].
- When [tex]\( x = 1 \)[/tex], [tex]\( y = 5 \)[/tex].
- When [tex]\( x = 2 \)[/tex], [tex]\( y = 25 \)[/tex].
- When [tex]\( x = 3 \)[/tex], [tex]\( y = 125 \)[/tex].
- When [tex]\( x = 4 \)[/tex], [tex]\( y = 625 \)[/tex].
- When [tex]\( x = 5 \)[/tex], [tex]\( y = 3125 \)[/tex].
2. Determine the relationship:
- Note that the value of [tex]\( y \)[/tex] seems to be a power of 5. Specifically, each [tex]\( y \)[/tex] value can be expressed as 5 raised to the power of the corresponding [tex]\( x \)[/tex] value.
- For each [tex]\( x \)[/tex], [tex]\( y = 5^x \)[/tex].
3. Verify the pattern:
- Calculate [tex]\( y \)[/tex] using [tex]\( y = 5^x \)[/tex] for each given [tex]\( x \)[/tex]:
- For [tex]\( x = 1 \)[/tex], [tex]\( y = 5^1 = 5 \)[/tex].
- For [tex]\( x = 2 \)[/tex], [tex]\( y = 5^2 = 25 \)[/tex].
- For [tex]\( x = 3 \)[/tex], [tex]\( y = 5^3 = 125 \)[/tex].
- For [tex]\( x = 4 \)[/tex], [tex]\( y = 5^4 = 625 \)[/tex].
- For [tex]\( x = 5 \)[/tex], [tex]\( y = 5^5 = 3125 \)[/tex].
The calculated [tex]\( y \)[/tex] values match those in the table perfectly, following the formula [tex]\( y = 5^x \)[/tex]. Therefore, the solution confirms the pattern and relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the given table.
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