Question

Prove the following identities. (a) F1 +F3 +F5 +...+F2n−1 = F2n. (b) F0 −F1 +F2 −F3 +...−F2n−1 +F2n = F2n−1 −1. (c) F02 +F12 +F2 +...+Fn2 = Fn ·Fn+1. (d) Fn−1Fn+1 − Fn2 = (−1)n.

Discrete math about Fibonacci numbers

Answer #1

Fibonacci Numbers.
The Fibonacci numbers are
1,1,2,3,5,8,13,21,….1,1,2,3,5,8,13,21,….
We can define them inductively by f1=1,f1=1, f2=1,f2=1, and
fn+2=fn+1+fnfn+2=fn+1+fn for n∈N.
Prove that fn=[(1+√5)n−(1−√5)n]/2n√5.

Recall that the Fibonacci numbers are deﬁned by F0 = 0,F1 = 1
and Fn+2 = Fn+1 + Fn for all n ∈N∪{0}.
(1) Make and prove an (if and only if) conjecture about which
Fibonacci numbers are multiples of 3.
(2) Make a conjecture about which Fibonacci numbers are
multiples of 2020. (You do not need to prove your conjecture.)
How many base cases would a proof by induction of your
conjecture require?

The Fibonacci numbers are defined recursively as follows: f0 =
0, f1 = 1 and fn = fn−1 + fn−2 for all n ≥ 2.
Prove that for all non-negative integers n: fn*fn+2 = ((fn+1))^
2 − (−1)^n

Let f1, f2, f3: [a,b] -->R be nonnegative concave functions
such that f1(a) = f2(a) = f3(a) = f1(b) = f2(b) = f3(b) = 0.
Suppose that max(f1) <= max(f2) <= max(f3).
Prove that: max(f1) + max(f2) <= max(f1+f2+f3)

The Fibonacci series is given by; F0=0, F1=1,F2=1,
F3=2,F4=3,…F(i)=F(i-1)+F(i-2)
Given that r^2=r+1. Show that F(i) ≥ r^{n-2}, where F(i) is the
i th element in the Fibonacci sequence

The Fibonacci sequence is defined as follows F0 = 0 and F1 = 1
with Fn = Fn−1 +Fn−2 for n > 1. Give the first five terms F0 −
F4 of the sequence. Then show how to find Fn in constant space Θ(1)
and O(n) time. Justify your claims

Solution.The Fibonacci numbers are defined by the recurrence
relation is defined F1 = 1, F2 = 1 and for n > 1, Fn+1 = Fn +
Fn−1. So the first few Fibonacci Numbers are: 1, 1, 2, 3, 5, 8, 13,
21, 34, 55, 89, 144, . . . There are numerous curious properties of
the Fibonacci Numbers Use the method of mathematical induction to
verify a: For all integers n > 1 and m > 0 Fn−1Fm + FnFm+1...

The Fibonacci series can be defined recursively
as: F1 = 0; F2 = 1; and
Fn = Fn-2 + Fn-1. Show inductively
that: (F1)2 + (F2)2 +
... + (Fn)2 =
(Fn)(Fn+1).

Three factories F1, F2 and F3 respectively produce 25%,
35% and 40% of the total number of electrical parts intended for
the assembly of a machine. These factories respectively produce 1%,
2% and 3% of defective parts.
We notice : The event A : "the part is produced by the F1
factory"
The event B : "the part is produced by the F2
factory"
The event C : "the part is produced by the F3 factory"
The event D : "the...

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.

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