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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, 4 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:):). · 2 years, 4 months ago

@Krishna Ar @Siddharth G · 2 years, 4 months ago

Perfect! I didn't go for x,y and faced problems in factorizing. PS: Small Typo at the end. · 2 years, 4 months ago

The substitution certainly helped made it easier to manipulate. Staff · 2 years, 4 months ago

Yes, however, can we do this with induction? · 2 years, 4 months ago

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 $$F_{n+1}$$. Staff · 2 years, 4 months ago

yes thanku!! · 2 years, 4 months ago

thanx!! · 2 years, 4 months ago

Absolutely right. This is wht I tried, but the $$X,Y$$ substitution, Mindblowing :D · 2 years, 4 months ago

thanx!! · 2 years, 4 months ago

How did you do? · 2 years, 4 months ago

Not well, 1b 5a, b 6b and half of the 8th were good. I left 1a half done. How was your paper? · 2 years, 4 months ago

Same problem with wet 3a. BTW are you sure that 6a is 'Yes'? · 2 years, 4 months ago

Oh Nono, I didnt understand it properly but gave some stupid explanation and put Yes. WBU? How did you do that question? · 2 years, 4 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. · 2 years, 4 months ago

Well, as I said...I wasnt even considering that question right :P. How was NTSE? (Range of marks..?) · 2 years, 4 months ago

Terrible! Expecting 114/140; expected cutoff~123 · 2 years, 4 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. · 2 years, 4 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. · 2 years, 4 months ago

Oh I see. Be optimistic though :) · 2 years, 4 months ago

Nah, just moving on. Are you giving RMO? · 2 years, 4 months ago

Yes I am. But NMTC has almost you know, shattered the vital mathematical force in me :3 · 2 years, 4 months ago

Oh, please. How was your NTSE? · 2 years, 4 months ago

Good. xD. It ough'to be that good lest you wouldn't qualify in TN. Expecting 130+, let's see.(the results) · 2 years, 4 months ago

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. :( · 2 years, 4 months ago