Certainly! Let's take a systematic approach to find the missing value in the table.
We have the following values in the table:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline 2 & 3 & 4 & 15 & 12 \\
\hline 3 & 4 & 5 & 28 & 20 \\
\hline 4 & 5 & 6 & 45 & 30 \\
\hline 5 & 6 & 7 & 66 & 42 \\
\hline 6 & 7 & 8 & ? & 56 \\
\hline
\end{array}
\][/tex]
We need to find the value for the question mark in the fourth column.
### Step 1: Determine the differences between the fourth column values
Let's start by examining the differences between consecutive values in the fourth column.
[tex]\[
\begin{align*}
28 - 15 &= 13 \\
45 - 28 &= 17 \\
66 - 45 &= 21 \\
\end{align*}
\][/tex]
### Step 2: Identify the pattern in differences
Now, let's examine the differences of these differences:
[tex]\[
\begin{align*}
17 - 13 &= 4 \\
21 - 17 &= 4 \\
\end{align*}
\][/tex]
We can see that the differences in the differences are constant ([tex]\(4\)[/tex]).
### Step 3: Predict the next difference
If the pattern continues, the next difference (between the unknown value and the last known value [tex]\(66\)[/tex]) should also be [tex]\(21 + 4 = 25\)[/tex].
### Step 4: Calculate the missing value
The missing value is obtained by adding the new difference to the last known value [tex]\(66\)[/tex]:
[tex]\[
66 + 25 = 91
\][/tex]
So, the completed table is:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline 2 & 3 & 4 & 15 & 12 \\
\hline 3 & 4 & 5 & 28 & 20 \\
\hline 4 & 5 & 6 & 45 & 30 \\
\hline 5 & 6 & 7 & 66 & 42 \\
\hline 6 & 7 & 8 & 91 & 56 \\
\hline
\end{array}
\][/tex]
Therefore, the missing value in the fourth column (where the question mark is) should be [tex]\( \boxed{91} \)[/tex].
