Problem 26

Solution 1
We begin filling in the table. The top row has a first term $1$ and a fifth term $25$, so we have the common difference is $\frac{25-1}4=6$. This means we can fill in the first row of the table:

The fifth row has a first term of $17$ and a fifth term of $81$, so the common difference is $\frac{81-17}4=16$. We can fill in the fifth row of the table as shown:

We must find the third term of the arithmetic sequence with a first term of $13$ and a fifth term of $49$. The common difference of this sequence is $\frac{49-13}4=9$, so the third term is $13+2\cdot 9=\boxed{\textbf{(B) }31}$.

Solution 2
The middle term of the first row is $\frac{25+1}{2}=13$, since the middle number is just the average in an arithmetic sequence. Similarly, the middle of the bottom row is $\frac{17+81}{2}=49$. Applying this again for the middle column, the answer is $\frac{49+13}{2}=\boxed{\textbf{(B)}~31}$.

Solution 3
The value of $X$ is simply the average of the average values of both diagonals that contain $X$. This is $\frac{\frac{1+81}{2}+\frac{17+25}{2}}{2} =\frac{\frac{82}{2}+\frac{42}{2}}{2} = \frac{41+21}{2} = \boxed{\textbf{(B)}~31}$