Difference between revisions of "2020 AIME II Problems/Problem 6"
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==Problem== | ==Problem== | ||
− | Define a sequence recursively by <math>t_1 = 20</math>, <math>t_2 = 21</math>, and<cmath>t_n = \frac{5t_{n-1}+1}{25t_{n-2}}</cmath>for all <math>n \ge 3</math>. Then <math>t_{2020}</math> can be | + | Define a sequence recursively by <math>t_1 = 20</math>, <math>t_2 = 21</math>, and<cmath>t_n = \frac{5t_{n-1}+1}{25t_{n-2}}</cmath>for all <math>n \ge 3</math>. Then <math>t_{2020}</math> can be expressed as <math>\frac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>. |
==Solution== | ==Solution== |
Revision as of 22:36, 24 September 2020
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:
Now using this information, as well as the previous information, we are able to determine the value of
:
Now using this information, as well as the previous information, we are able to determine the value of
:
Now using this information, as well as the previous information, we are able to determine the value of
:
Alas, we have figured this sequence is period 5! Thus, let us take , which is
, and therefore
. According to the original problem statement, our answer is essentially
. ~ nikenissan
Video Solution
https://youtu.be/_JTWJxbDC1A ~ CNCM
Video Solution 2
~IceMatrix
See Also
2020 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.