An arithmetic sequence is a sequence in which each term after the first is obtained by adding a constant to the previous term. For example, 2,4,6,8 and 1,4,7,10 are arithmetic sequences.

In the grid shown, the numbers in each row must form an arithmetic sequence and the numbers in each column must form an arithmetic sequence. The value of $x$ is
(A) 17
(B) 619
(C) 515
(D) 446
(E) 793

| 5 |  |  |  |
| :--- | :--- | :--- | :--- |
|  |  |  | 1211 |
|  |  | 1013 |  |
| 23 | $x$ |  |  |
We use labels $m$ and $n$ in the first column, as shown in the top grid.

Then, $5, m, n, 23$ are four terms of an arithmetic sequence.

Since 5 and 23 are three terms apart in this sequence, and their difference is $23-5=18$, the constant added to one term to obtain the next term in the first column is $\frac{18}{3}$ or 6 .

That is, $m=5+6=11$, and $n=11+6=17$.

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-217.jpg?height=314&width=311&top_left_y=160&top_left_x=1468)

(We confirm that the terms 5,11,17, 23 do form an arithmetic sequence.)

We use labels $p$ and $q$ in the second row, as shown in the middle grid.

Then, 11,p, $q, 1211$ are four terms of an arithmetic sequence. Since 11 and 1211 are three terms apart in this sequence, and their difference is $1211-11=1200$, the constant added to one term to obtain the next term in the second row is $\frac{1200}{3}$ or 400 . That is, $p=11+400=411$, and $q=411+400=811$.

(We confirm that the terms 11,411,811,1211 do form an

| 5 |  |  |  |
| ---: | ---: | ---: | ---: |
| 11 | $p$ | $q$ | 1211 |
| 17 |  | 1013 |  |
| 23 | $x$ |  |  |

arithmetic sequence.)

We use label $r$ in the third row, as shown in the third grid. Then, 17, $r, 1013$ are three terms of an arithmetic sequence. Since 17 and 1013 are two terms apart in this sequence, and their difference is $1013-17=996$, the constant added to one term to obtain the next term in the third row is $\frac{996}{2}$ or 498 . That is, $r=17+498=515$.

(We confirm that the terms 17,515, 1013 do form an arithmetic

| 5 |  |  |  |
| :---: | :---: | :---: | :---: |
| 11 | 411 | 811 | 1211 |
| 17 | $r$ | 1013 |  |
| 23 | $x$ |  |  |

sequence.)

Finally, in the second column the terms 411,515, $x$ are three terms of an arithmetic sequence.

Since 411 and 515 are one term apart in this sequence, the constant added to one term to obtain the next term in the second column is $515-411=104$.

Then, $x=515+104=619$.

The completed grid is as shown.

| 5 | 307 | 609 | 911 |
| ---: | :--- | :--- | ---: |
| 11 | 411 | 811 | 1211 |
| 17 | 515 | 1013 | 1511 |
| 23 | 619 | 1215 | 1811 |

ANSWER: (B)