Question

Please prove the following statement, in FULL detail. (by the if and only if proving technique, not induction!)

Prove that 5 | Un if and only if 5|n. Where Un is the Fibonacci sequence.

Answer #1

take mod5 and table it

Answer in FULL detail both proofs please!
Prove if 5 | Fn then 5 | n.
and then Prove
If 5 | n then 5 | Fn. Where Fn is the Fibonacci Numbers.

Please solve the following in FULL detail.
Using Un2 + (-1)n =
Un-1 * Un+1 to be true, Prove that any two
consecutive Fibonacci numbers are coprime.

Prove that 5 | Un if and only if 5|n. Where Un is the Fibonacci
sequence.

Please answer in FULL detail!!
Prove that the Mobius function is NOT
Completely Multiplicative. Where completely
multiplicative is μ(m*n) = μ(m) * μ(n).

This is more of a theoritical question: So, sometimes with
induction instead of proving the induction step (so n -> n+1)
its handier to prove the contrapositive of it. However, in cases
where this happens not only (not (n+1) -> not n) is being
proved, but also (not n -> not (n-1) ) etc until you prove the
contrapositive of the base case (it's called infinite descent). My
question is, why do you have to use infinite descent when proving...

Please answer the following in FULL detail!
Is there a Pythagorean triple consisting of three Fibonacci
numbers? Give an example if there is one, or a proof if there
isn't.

Prove by mathematical induction one of the following statements
:
a) 1 · 2 + 2 · 3 + 3 · 4 + . . . + n(n + 1) = n(n+1)(n+2) 3 for
all integer n ≥ 1.
b) u1 − u2 + u3 − u4 + . . . + (−1)n+1un = 1 + (−1)n+1un−1 for
all integer n ≥ 1. (un denotes the nth Fibonacci number)

Prove the following statement, please explain the steps,
m|n and n|m if and only if n = m or n = -m

3. Prove the following about the Fibonacci numbers:
(a) Fn is even if and only if n is divisible by 3.
(b) Fn is divisible by 3 if and only if n is divisible by 4.
(c) Fn is divisible by 4 if and only if n is divisible by 6.

Prove the following statement by mathematical induction. You
will not receive any credit unless you give a proof by induction.
Foreverypositiveintegern?Z+, wehave1+2+...+n=n(n+1)/2

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