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.

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.

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TopNewestLet 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 · 1 year, 11 months ago

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@Krishna Ar @Siddharth G – Adarsh Kumar · 1 year, 11 months ago

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x,yand faced problems in factorizing. PS: Small Typo at the end. – Siddharth G · 1 year, 11 months agoLog in to reply

– Calvin Lin Staff · 1 year, 11 months ago

The substitution certainly helped made it easier to manipulate.Log in to reply

– Siddharth G · 1 year, 11 months ago

Yes, however, can we do this with induction?Log in to reply

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 · 1 year, 11 months ago

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– Adarsh Kumar · 1 year, 11 months ago

yes thanku!!Log in to reply

– Adarsh Kumar · 1 year, 11 months ago

thanx!!Log in to reply

– Krishna Ar · 1 year, 11 months ago

Absolutely right. This is wht I tried, but the \(X,Y\) substitution, Mindblowing :DLog in to reply

– Adarsh Kumar · 1 year, 11 months ago

thanx!!Log in to reply

How did you do? – Krishna Ar · 1 year, 11 months ago

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– Siddharth G · 1 year, 11 months ago

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

– Siddharth G · 1 year, 11 months ago

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

– Krishna Ar · 1 year, 11 months ago

Oh Nono, I didnt understand it properly but gave some stupid explanation and put Yes. WBU? How did you do that question?Log in to reply

– Siddharth G · 1 year, 11 months ago

6a. Forthe transformation, we needed \(\Delta B=+2\) and \(\Delta O=0\). However with the allowed changes, these conditions were contradicting each other.Log in to reply

– Krishna Ar · 1 year, 11 months ago

Well, as I said...I wasnt even considering that question right :P. How was NTSE? (Range of marks..?)Log in to reply

– Siddharth G · 1 year, 11 months ago

Terrible! Expecting 114/140; expected cutoff~123Log in to reply

– Krishna Ar · 1 year, 11 months ago

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.Log in to reply

– Siddharth G · 1 year, 11 months ago

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.Log in to reply

– Krishna Ar · 1 year, 11 months ago

Oh I see. Be optimistic though :)Log in to reply

– Siddharth G · 1 year, 11 months ago

Nah, just moving on. Are you giving RMO?Log in to reply

– Krishna Ar · 1 year, 11 months ago

Yes I am. But NMTC has almost you know, shattered the vital mathematical force in me :3Log in to reply

– Siddharth G · 1 year, 11 months ago

Oh, please. How was your NTSE?Log in to reply

– Krishna Ar · 1 year, 11 months ago

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

Now , you clearly know I sucked more than you did. :( – Krishna Ar · 1 year, 11 months ago

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