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NMTC Problem 3a

The Fibonacci Sequence is defined by \(F_0 = 1\),\(F_1 =1\) and \(F_n=F_{n-1}+F_{n-2}\). Prove that \(7{F_{n+2}^3}-{F_n^3}-{F_{n+1}^3}\) is divisible by \(F_{n+3}\).

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

Note by Siddharth G
2 years, 6 months ago

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Let us take \(F_{n}=x\) and \(F_{n+1}=y \Rightarrow\ F_{n+2}=x+y \Rightarrow\ F_{n+3}=x+2y.\)Now,let us expand the given expression:\(: 7F_{n+2}^{3}-F_{n}^{3}-F_{n+1}^{3}\) in terms of \(x\) and \(y\).We get\(: 7(x+y)^{3}-x^{3}-y^{3}\).Simplifying that,we get\(: 6x^{3}+6y^{3}+21xy(x+y).\)Taking \((x+y)\) common we get\(: (x+y)(6x^{2}-6xy+6y^{2}+21xy) =(x+y)(6x^{2}+15xy+6y^{2}) =(x+y)(x+2y)(6x+3y).\)But,\((2y+x)=F_{n+3}.\)hence proved:):). Adarsh Kumar · 2 years, 6 months ago

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@Adarsh Kumar @Krishna Ar @Siddharth G Adarsh Kumar · 2 years, 6 months ago

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@Adarsh Kumar Perfect! I didn't go for x,y and faced problems in factorizing. PS: Small Typo at the end. Siddharth G · 2 years, 6 months ago

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@Siddharth G The substitution certainly helped made it easier to manipulate. Calvin Lin Staff · 2 years, 6 months ago

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@Calvin Lin Yes, however, can we do this with induction? Siddharth G · 2 years, 6 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 \( F_n \) doesn't tell you anything about divisibility by \( F_{n+1} \). Calvin Lin Staff · 2 years, 6 months ago

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@Siddharth G yes thanku!! Adarsh Kumar · 2 years, 6 months ago

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@Siddharth G thanx!! Adarsh Kumar · 2 years, 6 months ago

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@Adarsh Kumar Absolutely right. This is wht I tried, but the \(X,Y\) substitution, Mindblowing :D Krishna Ar · 2 years, 6 months ago

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@Krishna Ar thanx!! Adarsh Kumar · 2 years, 6 months ago

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How did you do? Krishna Ar · 2 years, 6 months ago

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@Krishna Ar Not well, 1b 5a, b 6b and half of the 8th were good. I left 1a half done. How was your paper? Siddharth G · 2 years, 6 months ago

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@Siddharth G Same problem with wet 3a. BTW are you sure that 6a is 'Yes'? Siddharth G · 2 years, 6 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 · 2 years, 6 months ago

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@Krishna Ar 6a. Forthe transformation, we needed \(\Delta B=+2\) and \(\Delta O=0\). However with the allowed changes, these conditions were contradicting each other. Siddharth G · 2 years, 6 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 · 2 years, 6 months ago

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@Krishna Ar Terrible! Expecting 114/140; expected cutoff~123 Siddharth G · 2 years, 6 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 · 2 years, 6 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 · 2 years, 6 months ago

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@Siddharth G Oh I see. Be optimistic though :) Krishna Ar · 2 years, 6 months ago

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@Krishna Ar Nah, just moving on. Are you giving RMO? Siddharth G · 2 years, 6 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 · 2 years, 6 months ago

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@Krishna Ar Oh, please. How was your NTSE? Siddharth G · 2 years, 6 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 · 2 years, 6 months ago

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@Siddharth G 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 · 2 years, 6 months ago

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