2020 AIME II Problems/Problem 6
Contents
Problem
Define a sequence recursively by ,
, and
for all
. Then
can be expressed as
, where
and
are relatively prime positive integers. Find
.
Solution
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
.
Solution 3
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
Video Solution
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 \frac{1}{5}, and t_5 can be rounded to \frac{1}{5}$. t_6 turns out to be around 20, which means that there is probably a loop with period 5. The rest of the solution proceeds normally.
~IceMatrix
See Also
2020 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
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