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Problem 596

AMC 10/12, early questions

Algebra Difficulty 3.8 Multiple choice CEMC Gauss (Grade 7) · Canada · 2013

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

Pick one

Official solution

We use labels, $m$ and $n$ , in the fourth row of the grid, as shown.

[[IMAGE0]]

Then, $10,m,36,n$ are four terms of an arithmetic sequence.

Since 10 and 36 are two terms apart in this sequence, and their difference is $36-10=26$ , the constant added to one term to obtain the next term in the fourth row is $\frac{26}{2}$ or 13.

That is, $m=10+13=23$ , and $n=36+13=49$ .

(We confirm that the terms $10,23,36,49$ do form an arithmetic sequence.)

In the fourth column, 25 and $n$ (which equals 49) are two terms apart in this sequence, and their difference is $49-25=24$ . Thus, the constant added to one term to obtain the next term in the fourth column is $\frac{24}{2}$ or 12.

That is, $x=25+12=37$ (or $x=49-12=37$ ).
The completed grid is as shown.

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.