2020 AIME II Problems/Problem 6
Define a sequence recursively by , , andfor all . Then can be expressed as , where and are relatively prime positive integers. Find .
Let . Then, we have where and . By substitution, we find , , , , and . So has a period of . Thus . So, . ~mn28407
Solution 2 (Official MAA)
More generally, let the first two terms be and and replace and in the recursive formula by and , respectively. Then some algebraic calculation shows that so the sequence is periodic with period . Therefore The requested sum is .
Let us examine the first few terms of this sequence and see if we can find a pattern. We are obviously given and , so now we are able to determine the numerical value of using this information:
Alas, we have figured this sequence is period 5! But since , we can state that . According to the original problem statement, our answer is . ~ nikenissan
https://youtu.be/_JTWJxbDC1A ~ CNCM
Video Solution 2
Quick way to notice recursion loop
Round the first two values to both be 20. Then, the next element can be rounded to . can then be quickly calculated to around , and can be rounded to . turns out to be around 20, which means that there is probably a loop with period 5. The rest of the solution proceeds normally.
|2020 AIME II (Problems • Answer Key • Resources)|
|1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15|
|All AIME Problems and Solutions|