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AoPS Volume 2, Problem 262
Shiyul 11
N
by Shiyul
Given that ![$\color[rgb]{0.35,0.35,0.35}v_1=2$](//latex.artofproblemsolving.com/2/0/4/204b13717fb1dec258e4917232ca9c96344031d2.png)
, ![$\color[rgb]{0.35,0.35,0.35}v_2=4$](//latex.artofproblemsolving.com/b/c/2/bc24e9589aaf75cb848e9fddeef43fb7a6fea2c3.png)
and ![$\color[rgb]{0.35,0.35,0.35} v_{n+1}=3v_n-v_{n-1}$](//latex.artofproblemsolving.com/8/7/3/8734d09c683e906771abf049d89b273196175585.png)
, prove that ![$\color[rgb]{0.35,0.35,0.35}v_n=2F_{2n-1}$](//latex.artofproblemsolving.com/b/1/e/b1efbac9eef93c5649dab70be54ae2286ca5b2f0.png)
, where the terms ![$\color[rgb]{0.35,0.35,0.35}F_n$](//latex.artofproblemsolving.com/0/2/5/025c95f57f5317da3d9739c1620448d8f7cf4e47.png)
are the Fibonacci numbers.
Can anyone give me hint on how to solve this (not solve the full problem). I'm not sure how to relate the v series to the Fibonacci sequence.
![$\color[rgb]{0.35,0.35,0.35}v_1=2$](http://latex.artofproblemsolving.com/2/0/4/204b13717fb1dec258e4917232ca9c96344031d2.png)
, ![$\color[rgb]{0.35,0.35,0.35}v_2=4$](http://latex.artofproblemsolving.com/b/c/2/bc24e9589aaf75cb848e9fddeef43fb7a6fea2c3.png)
and ![$\color[rgb]{0.35,0.35,0.35} v_{n+1}=3v_n-v_{n-1}$](http://latex.artofproblemsolving.com/8/7/3/8734d09c683e906771abf049d89b273196175585.png)
, prove that ![$\color[rgb]{0.35,0.35,0.35}v_n=2F_{2n-1}$](http://latex.artofproblemsolving.com/b/1/e/b1efbac9eef93c5649dab70be54ae2286ca5b2f0.png)
, where the terms ![$\color[rgb]{0.35,0.35,0.35}F_n$](http://latex.artofproblemsolving.com/0/2/5/025c95f57f5317da3d9739c1620448d8f7cf4e47.png)
are the Fibonacci numbers.Can anyone give me hint on how to solve this (not solve the full problem). I'm not sure how to relate the v series to the Fibonacci sequence.
11 replies
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Hard number theory
td12345 3
N
by mathprodigy2011
Let 
be a prime number. Define the set
![\[
M_q = \left\{ x \in \mathbb{Z}^* \,\middle|\, \sqrt{x^2 + 2q^{2025} x} \in \mathbb{Q} \right\}.
\]](//latex.artofproblemsolving.com/9/8/b/98b266a869cf5686cea86be51152ef234486993e.png)
Find the number of elements of
.
be a prime number. Define the set![\[
M_q = \left\{ x \in \mathbb{Z}^* \,\middle|\, \sqrt{x^2 + 2q^{2025} x} \in \mathbb{Q} \right\}.
\]](http://latex.artofproblemsolving.com/9/8/b/98b266a869cf5686cea86be51152ef234486993e.png)
Find the number of elements of
.
3 replies
![\[ \frac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)(58^4+324)}{(4^4+324)(16^4+324)(28^4+324)(40^4+324)(52^4+324)}. \]](http://latex.artofproblemsolving.com/c/f/0/cf007b7ce24aa95493617d1f4011e451e7c49126.png)
Hardest Computational Problem?
happypi31415 1
N
by mathprodigy2011
What do you guys think the hardest computational problem (for high school students) is?
1 reply
No bash for this inequality
giangtruong13 2
N
by giangtruong13
Let 
be positive real number satisfy that: 
.Find the minimum: 
be positive real number satisfy that: 
.Find the minimum: 
2 replies
Question abt directed angles
idk12345678 6
N
by idk12345678
If you have a diameter of a circle COA, and there is a point on the circle B, then how do you prove CBA is 90 degrees. Usually, i would use the inscribed angle theorem, but you cant divide directed angles by 2
6 replies
junior 3 and 4 var ineq (2019 Romanian NMO grade VII P1)
parmenides51 8
N
by Burak0609
a) Prove that for 
, holds 
b) Prove that for
with 
, holds 
, holds 
b) Prove that for
with 
, holds 
8 replies
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