E Show that the sum of the squares of the first n Fibonacci numbers is given by the formula u + už + už +...+u = u,un+1 [Hint: For n 2, u = Unun+1 - UnUn-1-] %3D
E Show that the sum of the squares of the first n Fibonacci numbers is given by the formula u + už + už +...+u = u,un+1 [Hint: For n 2, u = Unun+1 - UnUn-1-] %3D
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
Question 5
![S7 has p
(Hint: Use and the fact that = + 4(2"-²un-1).]
14, If un <b < Un+2 for n 4, that the a +b be
12. It was in are only five that are triangular
11. It can be that Un is by um (n > m), then the is a Fibonacci
the is to u 3n+2 =0 (mod 17). Because
mumber or um a both cases.
cording to Theorem 14.3, these are
1. Each of the remaining r
%3D
diVISIctors (there being only r primes in all). A contradiction occurs because
P13-149- 2221 has three prime factors.
%3D
m any prime p #3, 1t is known that either up-1 or u p+1 is divisible by p. Confirm
this in the cases of the primes 7, 11, 13, and 17
=1,2, ..., 10, show that Su+4(-1" is always a perfect square.
2.
1. Prove that if 2 |un, then 4 | (u2.-
For the Fibonacci sequence, establish the following:
u1); and similarly, if 3 | un, then 9|(u1-u1).
n+1
(a) un+3 = Un (mod 2), hence u3, u6, uo.
m Kaas = ... are all divisible by 5.
Show that the sum of the squares of the first n Fibonacci numbers is given by the formula
are all even integers.
..
3u, (mod 5), hence u5, u10, u15,
I+#n#n = "n+ - .· + En+n + n
[Hint: For n 2, u?
6. Utilize the identity in Problem 5 to prove that for n > 3
u1%3D3+3u1+2(u2+ u3+ ..+ už + u)
n-1
7. Evaluate gcd(ug, U12), gcd(u15, u20), and gcd(u24, U 36).
&. Find the Fibonacci numbers that divide both u24 and u36-
3. Use the fact that u u, 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 un
(d) 8 u, if and only if 6|n.
10. If gcd(m, n)
if and only if 5 |n.
for all m, n > 1.
1, prove that umun divides umn
%3D
number or
It was Um-r is a Fibonacci number. Give examples illustrating both cases.
numbers. Find them.
1, prove that 23-1un = n (mod 5).
%3D
· + u3n = 16!
+.
positive integer n for which
3D16!
\Hint: By Wil
integer.
Couation is equivalent to u3n+2 =0 (mod 17). Because](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F66c7b497-dac7-4855-b923-2e60bbc73063%2F379a8125-3802-4925-adb7-cf10826a2a12%2Fbhmrih.jpeg&w=3840&q=75)
Transcribed Image Text:S7 has p
(Hint: Use and the fact that = + 4(2"-²un-1).]
14, If un <b < Un+2 for n 4, that the a +b be
12. It was in are only five that are triangular
11. It can be that Un is by um (n > m), then the is a Fibonacci
the is to u 3n+2 =0 (mod 17). Because
mumber or um a both cases.
cording to Theorem 14.3, these are
1. Each of the remaining r
%3D
diVISIctors (there being only r primes in all). A contradiction occurs because
P13-149- 2221 has three prime factors.
%3D
m any prime p #3, 1t is known that either up-1 or u p+1 is divisible by p. Confirm
this in the cases of the primes 7, 11, 13, and 17
=1,2, ..., 10, show that Su+4(-1" is always a perfect square.
2.
1. Prove that if 2 |un, then 4 | (u2.-
For the Fibonacci sequence, establish the following:
u1); and similarly, if 3 | un, then 9|(u1-u1).
n+1
(a) un+3 = Un (mod 2), hence u3, u6, uo.
m Kaas = ... are all divisible by 5.
Show that the sum of the squares of the first n Fibonacci numbers is given by the formula
are all even integers.
..
3u, (mod 5), hence u5, u10, u15,
I+#n#n = "n+ - .· + En+n + n
[Hint: For n 2, u?
6. Utilize the identity in Problem 5 to prove that for n > 3
u1%3D3+3u1+2(u2+ u3+ ..+ už + u)
n-1
7. Evaluate gcd(ug, U12), gcd(u15, u20), and gcd(u24, U 36).
&. Find the Fibonacci numbers that divide both u24 and u36-
3. Use the fact that u u, 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 un
(d) 8 u, if and only if 6|n.
10. If gcd(m, n)
if and only if 5 |n.
for all m, n > 1.
1, prove that umun divides umn
%3D
number or
It was Um-r is a Fibonacci number. Give examples illustrating both cases.
numbers. Find them.
1, prove that 23-1un = n (mod 5).
%3D
· + u3n = 16!
+.
positive integer n for which
3D16!
\Hint: By Wil
integer.
Couation is equivalent to u3n+2 =0 (mod 17). Because
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