3. From Eq. (1), deduce that U2n-1 2. U2n 2. %D %3D n 2
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.6: Congruence Classes
Problem 25E
Related questions
Question
Question 3 14.3
![8. Prove integer can be as a sum of terms fr
Hint: = Un+1 + Un-2 = Un - and use Eq. (3)]
7. Represent the 50, 75, 100, and 125 as of Fibonacei numter
[Hint: By (c), = by Eq. (3) t
ELEMENTARY NUMBER THEORY
A Derive the following expression for the alternating sum of the first n>2F
numbers:
3. From Eq. (1), deduce that
U2n-1=
U2n =
4. Use the results of Problem 3 to obtain the following identities:
(a) u
2.
tu23D2u2n-1, n > 3.
(b) u+2+u1= 2(u; + u41), n > 2.
2(u + u1), n > 2.
2.
%3D
5. Establish that the formula
Un-1+(-1)"
%D
holds for n > 2 and use this to conclude that consecutive Fibonacci numbers a
relatively prime.
6. Without resorting to induction, derive the following identities:
(a) u -4u,nUn-1
[Hint: Start by squaring
(b) Un+1Un-1- un+2Un-2 = 2(-1)",n > 3.
[Hint: Put un+2 = Un+1 + un, Un-2 = Un
(c) u-un+2un-2 = (-1)", n 2 3.
[Hint: Mimic the proof of Eq. (3).]
(d) u- un+3un-3 = 4(-1)"+1, n 2 4.
n (p)
u?_2, n 2 3.
both un-2 = Un - Un-1
%3D
and u,n+l = Un + He-ld
and use
Un-1
Eq. (3).]
%3D
%3D
%3D
-n+2Un-2
%3D
%3D
z: By part (c), unt4u, =u?,,+ (-1)"+1, whereas by E4.
+2+(-1)+2,]
(e) UnUn+i4n+3zun+4 =u2 - 1, n 2 1.
%3D
with u, deleted).
(that is, the Fibonacci sequence
eauence u2, U3, U4,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F66c7b497-dac7-4855-b923-2e60bbc73063%2F6e7317e3-68e5-42f2-9096-6f0611f40c80%2F4w0103l.jpeg&w=3840&q=75)
Transcribed Image Text:8. Prove integer can be as a sum of terms fr
Hint: = Un+1 + Un-2 = Un - and use Eq. (3)]
7. Represent the 50, 75, 100, and 125 as of Fibonacei numter
[Hint: By (c), = by Eq. (3) t
ELEMENTARY NUMBER THEORY
A Derive the following expression for the alternating sum of the first n>2F
numbers:
3. From Eq. (1), deduce that
U2n-1=
U2n =
4. Use the results of Problem 3 to obtain the following identities:
(a) u
2.
tu23D2u2n-1, n > 3.
(b) u+2+u1= 2(u; + u41), n > 2.
2(u + u1), n > 2.
2.
%3D
5. Establish that the formula
Un-1+(-1)"
%D
holds for n > 2 and use this to conclude that consecutive Fibonacci numbers a
relatively prime.
6. Without resorting to induction, derive the following identities:
(a) u -4u,nUn-1
[Hint: Start by squaring
(b) Un+1Un-1- un+2Un-2 = 2(-1)",n > 3.
[Hint: Put un+2 = Un+1 + un, Un-2 = Un
(c) u-un+2un-2 = (-1)", n 2 3.
[Hint: Mimic the proof of Eq. (3).]
(d) u- un+3un-3 = 4(-1)"+1, n 2 4.
n (p)
u?_2, n 2 3.
both un-2 = Un - Un-1
%3D
and u,n+l = Un + He-ld
and use
Un-1
Eq. (3).]
%3D
%3D
%3D
-n+2Un-2
%3D
%3D
z: By part (c), unt4u, =u?,,+ (-1)"+1, whereas by E4.
+2+(-1)+2,]
(e) UnUn+i4n+3zun+4 =u2 - 1, n 2 1.
%3D
with u, deleted).
(that is, the Fibonacci sequence
eauence u2, U3, U4,
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