4.2 question 1
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
14.2 question 1
![[Hint: Use and the fact = +4(2"-2un-1).]
13. For n > 1, = n 5).
14. If un < a < Un+1 <b for n > 4, the a + b be
12. It was in 1989 that there are only are also triangular
11. It can be that when un is by um (n > m), then the is a Fibonacci
|-digit u604711.
of them are
. Pr, arranged in ascending order. Next, consider the correspond-
Gihonacci numbers u2, u3, u5.
in pairs. Exclude
elatively prime
divisible by a
U p.. According to Theorem 14.3, these are
1. Each of the remaining r – 1 numbers is
single prime with the possible exception that one of them has two
divisio stors (there being onlyr primes in all). A contradiction occurs because
U2
%3D
-
73- 149 · 2221 has three prime factors.
%3D
PROBLEMS 14.2
L Given any prime p # 5, it is known that either u „–1 or u p+1 is divisible by p. Confirm
this in the cases of the primes 7, 11, 13, and 17.
2. For n = 1, 2, ...,
3. Prove that if 2 |Un, then 4| (u1
4. For the Fibonacci sequence, establish the following:
(a) un+3 = Un (mod 2), hence u3, u6, u9, ... are all even integers.
(b) un+5 = 3un (mod 5), hence u5, u10, u15,
5. Show that the sum of the squares of the first n Fibonacci numbers is given by the formula
10, show that 5u, + 4(-1)" is always a perfect square.
- u²_1); and similarly, if 3 | un, then 9|(u41– u²-1).
are all divisible by 5.
..
+ u;
= Unun+1
I+unun = "n + ·..+ En + En + ¿n
[Hint: For n > 2, u = unun+1 – UnUn-1-]
0. Utilize the identity in Problem 5 to prove that for n > 3
[I-uun –
%3D
u41= u;+ 3u,-1 + 2(u,-2 + u;-3 + … ..+
Un+1
1. Evaluate gcd(u9, u12), gcd(u15, U20), and gcd(u24, u36).
and
U36.
3. Use the fact that u. Lun if and only if m |n to verify each of the assertions below:
(a) 2|u, if and only if 3 | n.
(b) 3| un if and only if 4 | n.
(c) 5| u,
(d) 8|un if and only if 6|n.
10. If gcd(m, n)
if and only if 5|n.
11. It can be shown that when un is divided by um (n > m), then the remainderr is a Fibonacci
number or um
1, prove that umun divides Umn for all m, n > 1.
%3D
n is a Fibonacci number. Give examples illustrating both cases.
numbers. Find them.
= 2(2"-lun) + 4(2"-2un-1).]
%3D
*0. For n > 1, prove that 2"-'un = n (mod 5).
number.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F66c7b497-dac7-4855-b923-2e60bbc73063%2F2f00c658-71a4-4c27-9f3f-07b381d9c7e3%2Fn12wzbz.jpeg&w=3840&q=75)
Transcribed Image Text:[Hint: Use and the fact = +4(2"-2un-1).]
13. For n > 1, = n 5).
14. If un < a < Un+1 <b for n > 4, the a + b be
12. It was in 1989 that there are only are also triangular
11. It can be that when un is by um (n > m), then the is a Fibonacci
|-digit u604711.
of them are
. Pr, arranged in ascending order. Next, consider the correspond-
Gihonacci numbers u2, u3, u5.
in pairs. Exclude
elatively prime
divisible by a
U p.. According to Theorem 14.3, these are
1. Each of the remaining r – 1 numbers is
single prime with the possible exception that one of them has two
divisio stors (there being onlyr primes in all). A contradiction occurs because
U2
%3D
-
73- 149 · 2221 has three prime factors.
%3D
PROBLEMS 14.2
L Given any prime p # 5, it is known that either u „–1 or u p+1 is divisible by p. Confirm
this in the cases of the primes 7, 11, 13, and 17.
2. For n = 1, 2, ...,
3. Prove that if 2 |Un, then 4| (u1
4. For the Fibonacci sequence, establish the following:
(a) un+3 = Un (mod 2), hence u3, u6, u9, ... are all even integers.
(b) un+5 = 3un (mod 5), hence u5, u10, u15,
5. Show that the sum of the squares of the first n Fibonacci numbers is given by the formula
10, show that 5u, + 4(-1)" is always a perfect square.
- u²_1); and similarly, if 3 | un, then 9|(u41– u²-1).
are all divisible by 5.
..
+ u;
= Unun+1
I+unun = "n + ·..+ En + En + ¿n
[Hint: For n > 2, u = unun+1 – UnUn-1-]
0. Utilize the identity in Problem 5 to prove that for n > 3
[I-uun –
%3D
u41= u;+ 3u,-1 + 2(u,-2 + u;-3 + … ..+
Un+1
1. Evaluate gcd(u9, u12), gcd(u15, U20), and gcd(u24, u36).
and
U36.
3. Use the fact that u. Lun if and only if m |n to verify each of the assertions below:
(a) 2|u, if and only if 3 | n.
(b) 3| un if and only if 4 | n.
(c) 5| u,
(d) 8|un if and only if 6|n.
10. If gcd(m, n)
if and only if 5|n.
11. It can be shown that when un is divided by um (n > m), then the remainderr is a Fibonacci
number or um
1, prove that umun divides Umn for all m, n > 1.
%3D
n is a Fibonacci number. Give examples illustrating both cases.
numbers. Find them.
= 2(2"-lun) + 4(2"-2un-1).]
%3D
*0. For n > 1, prove that 2"-'un = n (mod 5).
number.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
