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:
Let be the smallest number of apples and the common difference in the arithmetic sequence, with . Then the numbers of apples on the six trees are:
so their total is:
We are given:
The second equation simplifies to:
Substituting into the first equation:
so
Then:
The greatest number of apples on any tree is:
.
The problems on this page are the property of the MAA's American Mathematics Competitions