To determine the missing number in the sequence, we can start by examining the provided [tex]\( y \)[/tex]-values for [tex]\( x \)[/tex]-values to see if there is a pattern or sequence. The given data points are:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & 1 & 2 & 3 & 4 & 5 \\
\hline
y & 6 & 36 & 216 & ? ? ? & 7,776 \\
\hline
\end{array}
\][/tex]
We need to find the missing value for [tex]\( y \)[/tex] when [tex]\( x = 4 \)[/tex].
Let's examine the relationship between subsequent [tex]\( y \)[/tex]-values:
- [tex]\( \frac{y_2}{y_1} = \frac{36}{6} = 6 \)[/tex]
- [tex]\( \frac{y_3}{y_2} = \frac{216}{36} = 6 \)[/tex]
From this, it appears that the [tex]\( y \)[/tex]-values are forming a geometric progression with a common ratio of 6. If this pattern continues, then each [tex]\( y \)[/tex]-value is found by multiplying the previous [tex]\( y \)[/tex]-value by 6.
Therefore, to find the missing value at [tex]\( x = 4 \)[/tex]:
[tex]\[
y_{4} = y_{3} \times 6 = 216 \times 6
\][/tex]
Carrying out the multiplication:
[tex]\[
216 \times 6 = 1,296
\][/tex]
Thus, the missing number in the table is [tex]\( 1,296 \)[/tex].
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & 1 & 2 & 3 & 4 & 5 \\
\hline
y & 6 & 36 & 216 & 1{,}296 & 7{,}776 \\
\hline
\end{array}
\][/tex]
Hence, the missing number in the sequence, which is consistent with the given choices, is 1,296.
