2023 AIME II Problems/Problem 1
Contents
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 1
In the arithmetic sequence, let be the first term and be the common difference, where The sum of the first six terms is We are given that The second equation implies that Substituting this into the first equation, we get It follows that Therefore, the greatest number of apples growing on any of the six trees is
~MRENTHUSIASM
Solution 2
Let the terms in the sequence be defined as
Since this is an arithmetic sequence, we have So, Hence, And, since we are given that we get and
~Kiran
Video Solution 1 by SpreadTheMathLove
https://www.youtube.com/watch?v=nNhfDCX5-bw
Video Solution by the Power of Logic(both #1 and #2)
See also
2023 AIME II (Problems • Answer Key • Resources) | ||
Preceded by First Problem |
Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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