Problem:
The numbers of apples growing on each of six apple trees form an arithmetic sequence where the greatest number of apples growing on any of the six trees is double the least number of apples growing on any of the six trees. The total number of apples growing on all six trees is . Find the greatest number of apples growing on any of the six trees.
Solution:
Answer (220):
Because the numbers of apples growing on the six trees form an arithmetic sequence, there are positive integers and such that the numbers of apples are , and . This is a total of apples. Also, the greatest number of apples growing on any of the six trees is , and this is double the least number of apples growing on any of the six trees, which is . Thus , so . This means , and . The greatest number of apples growing on any of the trees is double this, which is 220 .
OR
Let be the greatest number of apples growing on any of the trees. Then the least number of apples growing on any of the trees is . Because the numbers of apples growing on the trees form an arithmetic sequence with 6 terms, it follows that the total number of apples growing on the trees is . The greatest number of apples growing on any of the trees is therefore .
The problems and solutions on this page are the property of the MAA's American Mathematics Competitions