To find the missing number in the table where [tex]\( x \)[/tex] increases and [tex]\( y \)[/tex] follows a certain pattern, let's determine the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
Given [tex]\( x \)[/tex] values:
[tex]\[ x = 1, 2, 3, 4, 5 \][/tex]
The corresponding [tex]\( y \)[/tex] values are:
[tex]\[ y = 6, 36, 216, ???, \text{7,776} \][/tex]
Observing the given values:
- When [tex]\( x = 1 \)[/tex], [tex]\( y = 6 \)[/tex]
- When [tex]\( x = 2 \)[/tex], [tex]\( y = 36 \)[/tex]
- When [tex]\( x = 3 \)[/tex], [tex]\( y = 216 \)[/tex]
- When [tex]\( x = 5 \)[/tex], [tex]\( y = 7,776 \)[/tex]
To determine the pattern for [tex]\( y \)[/tex], let's find out how the value of [tex]\( y \)[/tex] changes based on [tex]\( x \)[/tex].
Let's analyze the relationship:
1. For [tex]\( x = 1 \)[/tex]:
[tex]\[ y = 6 \][/tex]
2. For [tex]\( x = 2 \)[/tex]:
[tex]\[ y = 36 \][/tex]
3. For [tex]\( x = 3 \)[/tex]:
[tex]\[ y = 216 \][/tex]
4. For [tex]\( x = 4 \)[/tex]:
[tex]\[ y = ??? \][/tex]
5. For [tex]\( x = 5 \)[/tex]:
[tex]\[ y = 7,776 \][/tex]
We need to deduce [tex]\( y \)[/tex] when [tex]\( x = 4 \)[/tex].
By observing the pattern in [tex]\( y \)[/tex], specifically focusing on the intervals in multipliers or factorial growth:
If we consider the function:
[tex]\[ y = 6 \times x! \][/tex]
Let's test this hypothesis:
1. For [tex]\( x = 1 \)[/tex]:
[tex]\[ y = 6 \times 1! = 6 \][/tex]
2. For [tex]\( x = 2 \)[/tex]:
[tex]\[ y = 6 \times 2! = 6 \times 2 = 12 \][/tex]
[tex]\[ y = 36 \text{ (requires additional multiplication factor)} \][/tex]
To ease calculations, note an extra multiplier. It should appear factorial related:
Find [tex]\( x = 3 \)[/tex]'s reliable projection:
[tex]\[ y = 6 \times 3! = 6 \times 6 = 36 \text{ (requires another factor, conclusion multiplied)} \][/tex]
Using reviewed elements directly:
When [tex]\( x = 4 \)[/tex]:
[tex]\[ y = 4 \times 4! \times 6 \][/tex]
Calculate factorial [tex]\( 4! \)[/tex]:
[tex]\[ 4! = 4 \times 3 \times 2 = 24 \][/tex]
Incorporating [tex]\( x \)[/tex]:
[tex]\[ y = 4 \times 24 \times 6 = 576 \][/tex]
Thus, the missing value in the table for [tex]\( x = 4 \)[/tex] is:
[tex]\[ \boxed{576} \][/tex]
