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?
. Fibonacci sequence Fn is defined as follows: F1 = 1; F2 = 1;
Fn = Fn−1 + Fn−2, ∀n ≥ 3. A pseudocode is given below which returns
n’th number in the Fibonacci sequence. RecursiveFibonacci(n) 1 if n
= 0 2 return 0 3 elseif n = 1 or n = 2 4 return 1 5 else 6 a =
RecursiveFibonacci(n − 1) 7 b = RecursiveFibonacci(n − 2) 8 return
a + b Derive the complexity of the...
0.3 The Fibonacci numbers Fn are defined by F1 = 1, F2 = 1 and
for n >2, Fn = F sub (n-1) + F sub (n-2). Find a formula for Fn
by solving the difference equation.
For each n ∈ N, let fn : [0, 1] → [0, 1] be defined
by fn(x) = 0, x > 1/n and fn(x) = 1−nx if
0 ≤ x ≤1/n.
The collection {fn(x) : n ∈ N} converges to a point,
call it f(x) for each x ∈ [0, 1]. Show whether {fn(x) :
n ∈ N}
converges to f uniformly or not.
The Fibonacci sequence is defined as F_1 = 1, F_2 = 1, and F_n =
F_n-1 + F_n-2 for n >= 3. Calculate the sum F_1 + F_2 + ... +
F_n using the fundamental theorem of summation.
Let’s revisit the Fibonacci sequence, this time without an
array. Recall that the sequence is defined by the following
recurrence relation: F0 = 0 F1 = 1 Fk = Fk-1 + Fk-2 So, the
sequence looks like: 0, 1, 1, 2, 3, 5, 8, 13, 21, … Write a program
fib.c consisting only of a main() function that prompts the user
for the last Fibonacci number n to print, and then prints them from
F0 to Fn. (No file input...
2. The Fibonacci sequence is defined as
f(n) = f(n - 1) + f(n - 2)
with f(0) = 0 and f(1) = 1.
Find f(54) by a program or maually. Note that this number must
be positive
and f(53) = 53.......73 (starting with 53 and ending with 73).
I must admit that
my three machines including a desktop are unable to find f(54)
and they
quit during computation.
The answer is f(54) = 86267571272
*/
The Java code:
public...
Let {an} be a sequence defined recursively by a1 = 1 and an+1 =
2√ 1 + an where n ∈ N
(b) Does {an} converge or diverge? Justify your answer, making
sure to cite appropriate hypotheses/theorem(s) used. Hint : Try
BMCT [WHY?].
(c) (Challenge) If {an} converges then find its limit. Make sure
to fully justify your answer.