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.

No vote yet

1 vote

×

Problem Loading...

Note Loading...

Set Loading...

Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

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:):).

Log in to reply

@Krishna Ar @Siddharth G

Log in to reply

Perfect! I didn't go for

x,yand faced problems in factorizing. PS: Small Typo at the end.Log in to reply

Log in to reply

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 \( F_{n+1} \).

Log in to reply

Log in to reply

Log in to reply

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

Log in to reply

Log in to reply

How did you do?

Log in to reply

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

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

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

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. :(

Log in to reply