NMTC Problem 3a

The Fibonacci Sequence is defined by F0=1F_0 = 1,F1=1F_1 =1 and Fn=Fn1+Fn2F_n=F_{n-1}+F_{n-2}. Prove that 7Fn+23Fn3Fn+137{F_{n+2}^3}-{F_n^3}-{F_{n+1}^3} is divisible by Fn+3F_{n+3}.

This a part of my set NMTC 2nd Level (Junior) held in 2014.

Note by Siddharth G
4 years, 11 months ago

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Let us take Fn=xF_{n}=x and Fn+1=y Fn+2=x+y Fn+3=x+2y.F_{n+1}=y \Rightarrow\ F_{n+2}=x+y \Rightarrow\ F_{n+3}=x+2y.Now,let us expand the given expression::7Fn+23Fn3Fn+13: 7F_{n+2}^{3}-F_{n}^{3}-F_{n+1}^{3} in terms of xx and yy.We get:7(x+y)3x3y3: 7(x+y)^{3}-x^{3}-y^{3}.Simplifying that,we get:6x3+6y3+21xy(x+y).: 6x^{3}+6y^{3}+21xy(x+y).Taking (x+y)(x+y) common we get:(x+y)(6x26xy+6y2+21xy)=(x+y)(6x2+15xy+6y2)=(x+y)(x+2y)(6x+3y).: (x+y)(6x^{2}-6xy+6y^{2}+21xy) =(x+y)(6x^{2}+15xy+6y^{2}) =(x+y)(x+2y)(6x+3y).But,(2y+x)=Fn+3.(2y+x)=F_{n+3}.hence proved:):).

Adarsh Kumar - 4 years, 11 months ago

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@Krishna Ar @Siddharth G

Adarsh Kumar - 4 years, 11 months ago

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Absolutely right. This is wht I tried, but the X,YX,Y substitution, Mindblowing :D

Krishna Ar - 4 years, 11 months ago

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@Krishna Ar thanx!!

Adarsh Kumar - 4 years, 11 months ago

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Perfect! I didn't go for x,y and faced problems in factorizing. PS: Small Typo at the end.

Siddharth G - 4 years, 11 months ago

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@Siddharth G thanx!!

Adarsh Kumar - 4 years, 11 months ago

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@Siddharth G yes thanku!!

Adarsh Kumar - 4 years, 11 months ago

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@Siddharth G The substitution certainly helped made it easier to manipulate.

Calvin Lin Staff - 4 years, 11 months ago

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@Calvin Lin Yes, however, can we do this with induction?

Siddharth G - 4 years, 11 months ago

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@Siddharth G Possibly, but I don't see why that would work, nor do I see a way to start.

There is very little here to motivate a solution by induction. Knowing divisibility by Fn F_n doesn't tell you anything about divisibility by Fn+1 F_{n+1} .

Calvin Lin Staff - 4 years, 11 months ago

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How did you do?

Krishna Ar - 4 years, 11 months ago

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Not well, 1b 5a, b 6b and half of the 8th were good. I left 1a half done. How was your paper?

Siddharth G - 4 years, 11 months ago

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Sigh, atleast you had one full question to your credit. I got the first one fully , messed up second one (after getting half of it), third one wasn't salubrious, (I tried both, got to almost the answers), 4th I didnt do, 5a I didnt know, 5 b I got it, 6- I wrote Yes and No alternatingly :P, 7th I almost got it but lost it due to calculation error (You wont believe I put 40 cubed= 16000 :( )..8th I am not sure of it's accuracy....

Now , you clearly know I sucked more than you did. :(

Krishna Ar - 4 years, 11 months ago

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Same problem with wet 3a. BTW are you sure that 6a is 'Yes'?

Siddharth G - 4 years, 11 months ago

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@Siddharth G Oh Nono, I didnt understand it properly but gave some stupid explanation and put Yes. WBU? How did you do that question?

Krishna Ar - 4 years, 11 months ago

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@Krishna Ar 6a. Forthe transformation, we needed ΔB=+2\Delta B=+2 and ΔO=0\Delta O=0. However with the allowed changes, these conditions were contradicting each other.

Siddharth G - 4 years, 11 months ago

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@Siddharth G Well, as I said...I wasnt even considering that question right :P. How was NTSE? (Range of marks..?)

Krishna Ar - 4 years, 11 months ago

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@Krishna Ar Terrible! Expecting 114/140; expected cutoff~123

Siddharth G - 4 years, 11 months ago

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@Siddharth G Oh, I am sorry for bringing that up. How did you predict the cutoff so soon? (Institute?)...And, Could you tell me the number of seats for general cat. in Delhi.

Krishna Ar - 4 years, 11 months ago

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@Krishna Ar Not really, mostly from the marks of competitive peers. The no. of seats was 60 last year, however, I think they are going to increase it. FIITJEE predicts it to be near 119 due to this.

Siddharth G - 4 years, 11 months ago

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@Siddharth G Oh I see. Be optimistic though :)

Krishna Ar - 4 years, 11 months ago

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@Krishna Ar Nah, just moving on. Are you giving RMO?

Siddharth G - 4 years, 11 months ago

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@Siddharth G Yes I am. But NMTC has almost you know, shattered the vital mathematical force in me :3

Krishna Ar - 4 years, 11 months ago

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@Krishna Ar Oh, please. How was your NTSE?

Siddharth G - 4 years, 11 months ago

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@Siddharth G Good. xD. It ough'to be that good lest you wouldn't qualify in TN. Expecting 130+, let's see.(the results)

Krishna Ar - 4 years, 11 months ago

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