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.

#Algebra #RecurrenceRelations #FibonacciNumbers #NMTC

Easy Math Editor

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Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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 - 6 years, 7 months ago

@Krishna Ar @Siddharth G

Adarsh Kumar - 6 years, 7 months ago

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

Krishna Ar - 6 years, 7 months ago

@Krishna Ar – thanx!!

Adarsh Kumar - 6 years, 7 months ago

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

Siddharth G - 6 years, 7 months ago

@Siddharth G – thanx!!

Adarsh Kumar - 6 years, 7 months ago

@Siddharth G – yes thanku!!

Adarsh Kumar - 6 years, 7 months ago

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

Calvin Lin Staff - 6 years, 7 months ago

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

Siddharth G - 6 years, 7 months ago

@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 - 6 years, 6 months ago

How did you do?

Krishna Ar - 6 years, 7 months ago

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

Siddharth G - 6 years, 7 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. :(

Krishna Ar - 6 years, 7 months ago

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

Siddharth G - 6 years, 7 months ago

@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 - 6 years, 7 months ago

@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 - 6 years, 7 months ago

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

Krishna Ar - 6 years, 7 months ago

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

Siddharth G - 6 years, 7 months ago

@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 - 6 years, 7 months ago

@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 - 6 years, 7 months ago

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

Krishna Ar - 6 years, 7 months ago

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

Siddharth G - 6 years, 7 months ago

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

Krishna Ar - 6 years, 7 months ago

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

Siddharth G - 6 years, 7 months ago

@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 - 6 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$