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 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).

The following is Algorithm 8 from §5.4. Note that it uses the
following definition of the fibonacci sequence: fn = fn−1 +
fn−2, f1 = 1, f0 = 0.
procedure iterative fibonacci(n: nonnegative integer)
if n = 0 then return 0
else
x := 0 y := 1
for i := 1 to n − 1 do
z := x + y x := y y := z
end for
return y
end if
end procedure
Prove this algorithm is...

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...

Consider the following functions.
f1(x) = x, f2(x) = x-1, f3(x) = x+4
g(x) = c1f1(x) + c2f2(x) + c3f3(x)
Solve for c1, c2, and c3 so that g(x) = 0 on the interval (−∞, ∞).
If a nontrivial solution exists, state it. (If only the trivial
solution exists, enter the trivial solution {0, 0, 0}.)
{c1, c2, c3} =?
Determine whether f1, f2, f3 are linearly independent on the
interval (−∞, ∞).
linearly dependent or linearly independent?

Problem Two You are considering two mutually exclusive projects
with the following cash flows:
Project C/F0 C/F1 C/F2 C/F3 C/F4 C/F5 C/F6
A $(41,215) $12,500 $14,000 $16,500 $18,000 $20,000 N/A
B $(46,775) $15,000 $15,000 $15,000 $15,000 $15,000 $15,000
A) Assuming that the discount rate for project A is 16% and the
discount rate for B is 15%, then given that these are mutually
exclusive projects, which project would you take and why?
B) If you are one of the management...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 23 minutes ago

asked 23 minutes ago

asked 28 minutes ago

asked 28 minutes ago

asked 28 minutes ago

asked 35 minutes ago

asked 44 minutes ago

asked 45 minutes ago

asked 52 minutes ago

asked 53 minutes ago

asked 56 minutes ago

asked 56 minutes ago