Complete the table:

\begin{tabular}{|l|l|l|l|l|}
\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 & [tex][tex]$?$[/tex][/tex] & 56 \\
\hline
\end{tabular}



Answer :

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].