P. C 12...., 10, show that 5u, + 4(-1)“ is always a perfect square. 2. For n = 3. Prove that if 2|un, then 4| (u? For the Fibonacci sequence, establish the following: ,2 u- +1– u-1). 1); and similarly, if 3 | un, then 9 | (u - n+1
P. C 12...., 10, show that 5u, + 4(-1)“ is always a perfect square. 2. For n = 3. Prove that if 2|un, then 4| (u? For the Fibonacci sequence, establish the following: ,2 u- +1– u-1). 1); and similarly, if 3 | un, then 9 | (u - n+1
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.2: Exponents And Radicals
Problem 90E
Related questions
Question
3
![(Hint: Use fact that 2" = + 4(2"-2Un-1).]
12. It was in 1989 that there are only five are also triangul=
11. It can be that when un is divided by um (n > m), then the is a Fibonacc
-r is a Give both cases.
CI NUMBERS
are somewhat sparse; only 42 of them are
being the 126377-digit u604711.
present
Pr, arranged in ascending order. Next, consider the correspond-
U5,
U p, . According to Theorem 14.3, these are
1. Each of the remaining
..
primes 2, 3, 5.
r - 1 numbers is
%3D
Up-1 or u p+1
divisible by p. Confirm
PROBLEMS 14.2
u-).
is in the cases of the primes 7, 11, 13, and 17.
n+1
%3D
ST
are all even integers.
are all divisible by 5.
Prove that if 2 |Un, then 4 | (un+1 -Un-1); and similarly, if 3 | un, then 9 | (u²1 – u? ,).
(a) un+3 = Un (mod 2), hence u3, u6, U9,
(b) un15 = 3un (mod 5), hence u5, u10, u15,
E Show that the sum of the squares of the first n Fibonacci numbers is given by the formula
3. Prove that if 2|Un, then 4| (u?
..
%3D
Itun"n = "n+
..+ En + Gn + ;n
[Hint: For n > 2, u = unun+1 - unun-1.]
6. Utilize the identity in Problem 5 to prove that for n > 3
= u+ 3u-1 + 2(u, 2 +
ude
%3D
7. Evaluate gcd(u9, u12), gcd(u15, u20), and gcd(u24, U 36).
8. Find the Fibonacci numbers that divide both u24 and u 36.
u= u;+3u-1 +2(u-2 + u-3 + ·..+ už + uf)
. Use the fact that u m lu, if and only if m | n to verify each of the assertions below:
(a) 2|un if and only if 3 | n.
Un+1
arize
3.
= (21
(1)
(b) 3| un if and only if 4 | n.
(c) 5| un if and only if 5 | n.
(d) 8| un if and only if 6 | n.
%3D
for all m, n > 1.
divides umn
%3D
number or u m
Is a Fibonacci number. Give examples illustrating both cases.
numbers. Find them.
%3D
Hin , prove that 2"-lun =n (mod 5).
a Fibon](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F66c7b497-dac7-4855-b923-2e60bbc73063%2Fff121f14-a5a7-4abd-82bb-8601b1c1c28d%2Fd34sje.jpeg&w=3840&q=75)
Transcribed Image Text:(Hint: Use fact that 2" = + 4(2"-2Un-1).]
12. It was in 1989 that there are only five are also triangul=
11. It can be that when un is divided by um (n > m), then the is a Fibonacc
-r is a Give both cases.
CI NUMBERS
are somewhat sparse; only 42 of them are
being the 126377-digit u604711.
present
Pr, arranged in ascending order. Next, consider the correspond-
U5,
U p, . According to Theorem 14.3, these are
1. Each of the remaining
..
primes 2, 3, 5.
r - 1 numbers is
%3D
Up-1 or u p+1
divisible by p. Confirm
PROBLEMS 14.2
u-).
is in the cases of the primes 7, 11, 13, and 17.
n+1
%3D
ST
are all even integers.
are all divisible by 5.
Prove that if 2 |Un, then 4 | (un+1 -Un-1); and similarly, if 3 | un, then 9 | (u²1 – u? ,).
(a) un+3 = Un (mod 2), hence u3, u6, U9,
(b) un15 = 3un (mod 5), hence u5, u10, u15,
E Show that the sum of the squares of the first n Fibonacci numbers is given by the formula
3. Prove that if 2|Un, then 4| (u?
..
%3D
Itun"n = "n+
..+ En + Gn + ;n
[Hint: For n > 2, u = unun+1 - unun-1.]
6. Utilize the identity in Problem 5 to prove that for n > 3
= u+ 3u-1 + 2(u, 2 +
ude
%3D
7. Evaluate gcd(u9, u12), gcd(u15, u20), and gcd(u24, U 36).
8. Find the Fibonacci numbers that divide both u24 and u 36.
u= u;+3u-1 +2(u-2 + u-3 + ·..+ už + uf)
. Use the fact that u m lu, if and only if m | n to verify each of the assertions below:
(a) 2|un if and only if 3 | n.
Un+1
arize
3.
= (21
(1)
(b) 3| un if and only if 4 | n.
(c) 5| un if and only if 5 | n.
(d) 8| un if and only if 6 | n.
%3D
for all m, n > 1.
divides umn
%3D
number or u m
Is a Fibonacci number. Give examples illustrating both cases.
numbers. Find them.
%3D
Hin , prove that 2"-lun =n (mod 5).
a Fibon
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