3. From Eq. (1), deduce that U2n-1 2. U2n 2. %D %3D n 2

Advanced Engineering Mathematics
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Chapter2: Second-order Linear Odes
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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,
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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