where 
is a natural number.What is the remainder when 
be real numbers so that 
and 
Show that
![$\color[rgb]{0.35,0.35,0.35}v_1=2$](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)
,![$\color[rgb]{0.35,0.35,0.35}v_n=2F_{2n-1}$](http://latex.artofproblemsolving.com/b/1/e/b1efbac9eef93c5649dab70be54ae2286ca5b2f0.png)
,![$\color[rgb]{0.35,0.35,0.35}F_n$](http://latex.artofproblemsolving.com/0/2/5/025c95f57f5317da3d9739c1620448d8f7cf4e47.png)
are the Fibonacci numbers.
be real numbers . Prove that
![$$-\frac{1}{30}\leq \frac{ab+a+b+4}{(a^2+a+5)(b^2+b+5)}\leq \frac{64+24\sqrt[3] 6+9\sqrt[3] {36} }{700} $$](http://latex.artofproblemsolving.com/d/a/2/da21df4e61f8934aa355a68cae4d3e274c176741.png)
Y by Adventure10, Mango247
A rectangular sheet of paper (white on one side and gray on the other) was folded three times, as shown in the figure:
Rectangle
, which was white after the first fold, has 
cm more perimeter than rectangle 
, which was white after the second fold, and this in turn has 
cm more perimeter than rectangle 
, which was white after the third fold. Determine the area of the sheet.
Rectangle
, which was white after the first fold, has 
cm more perimeter than rectangle 
, which was white after the second fold, and this in turn has 
cm more perimeter than rectangle 
, which was white after the third fold. Determine the area of the sheet.This post has been edited 4 times. Last edited by parmenides51, Jun 8, 2024, 5:33 PM
,
.
and 
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