Library / /350 of 371

, 2013

Algebra Difficulty 3.8 AMC 10/12 Find the answer Canada

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

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.

Figure for this problem

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

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