math history---sequences?

Question

If Un<---(suppose to read U sub n) represents the nth term of the Fibonacci sequence show that, Un and Un+1<---(suppose to read U sub n+1) are relatively prime. okay, I dont knoq how to start this, and what exactly does the "sub n and sub n+1" do to the U in this sequence I dont see a relation, because the sequence is 1,1, 2, 3, 5, 8,..., x, y, x+y, etc. How do I show these are prime by using what they are asking.

Answer 1

Two integers are "relatively prime" if they have no factors in common other than 1 (or -1). Alternatively, you could say that j and k are relatively prime if, for any common divisor p of j and k, p also divides 1.

Well, suppose some factor p divided both the nth and (n+1)st term. The (n-1)st term is just the difference of the next two. And if p divides two numbers, it divides their difference. That can be worked all the backward until you get to the first terms in the sequence, and p divides 1.

To make this super-rigorous, you could say "Let n be the SMALLEST n such that the nth and n+1st terms share a common factor other than 1 and -1", and quickly derive a contradiction like above. You'd also have to check that n isn't 1, technically -- which makes this a WHOLE lot like a proof-by-induction.

And I'm almost certain the above is a lot more than you actually wanted to know. ;)

Answer 2

Fibonacci Number is given by
for n = 0 and 1, F(n) =1
for n > 1, F(n) = F(n -1) + F(n -2)

when
n = 0; F(0) =1
n = 1; F(1) = 1
n = 2; F(2) = F(1) + F(0) = 1 + 1= 2
n = 3; F(3) = F(3) + F(1) = 2 + 1= 3
n = 4; F(4) = F(3) + F(3) = 3 + 2= 5

Answer 3

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