295 For any primep 2 or 5, it is known that either Fp-1 or Fp+1 is divisible by p. Confirm this in the case of the primes 7, 11, 13, and 17. 3. From the formula F+ F- F2 (-1)", conclude that consecutive Fibonacci numbers are relatively prime. 4. n One can prove that the greatest common divisor of two Fibonacci numbers is also a Fibonacci number; 5. specifically, where d gcd (n, m). gcd (F1, Fm)= Fd no ner. Verify this identity in the case of gcd(F9, F12) and gcd(F15, F20). mula Use Problem 5 to prove that for n > 2, Fn| Fm if and only if n m. 6. no2 7. Establish each of the following assertions: 2|F, (that is, F, is even) if and only if 3 n. (b) 31F, if and only if 4|n. (c) 4|F, if and only if 6 n. (d) 5F, if and only if 5 n. [Hint: All these require the aid of the previous problem.] 8. Show that the sum of the squares of the first n Fibonacci numbers is given by the formula ing F2+F+F. .+ F= F,F+ 1 [Hint: Note that F F,(Fn+1 - F1-1) F,F +1- FrFn-1.]
295 For any primep 2 or 5, it is known that either Fp-1 or Fp+1 is divisible by p. Confirm this in the case of the primes 7, 11, 13, and 17. 3. From the formula F+ F- F2 (-1)", conclude that consecutive Fibonacci numbers are relatively prime. 4. n One can prove that the greatest common divisor of two Fibonacci numbers is also a Fibonacci number; 5. specifically, where d gcd (n, m). gcd (F1, Fm)= Fd no ner. Verify this identity in the case of gcd(F9, F12) and gcd(F15, F20). mula Use Problem 5 to prove that for n > 2, Fn| Fm if and only if n m. 6. no2 7. Establish each of the following assertions: 2|F, (that is, F, is even) if and only if 3 n. (b) 31F, if and only if 4|n. (c) 4|F, if and only if 6 n. (d) 5F, if and only if 5 n. [Hint: All these require the aid of the previous problem.] 8. Show that the sum of the squares of the first n Fibonacci numbers is given by the formula ing F2+F+F. .+ F= F,F+ 1 [Hint: Note that F F,(Fn+1 - F1-1) F,F +1- FrFn-1.]
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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#3 please
![295
For any primep 2 or 5, it is known that either Fp-1
or Fp+1 is divisible by p. Confirm this in the case of
the primes 7, 11, 13, and 17.
3.
From the formula F+ F- F2 (-1)", conclude
that consecutive Fibonacci numbers are relatively
prime.
4.
n
One can prove that the greatest common divisor of two
Fibonacci numbers is also a Fibonacci number;
5.
specifically,
where d
gcd (n, m).
gcd (F1, Fm)= Fd
no
ner.
Verify this identity in the case of gcd(F9, F12) and
gcd(F15, F20).
mula
Use Problem 5 to prove that for n > 2, Fn| Fm if and
only if n m.
6.
no2
7. Establish each of the following assertions:
2|F, (that is, F, is even) if and only if 3 n.
(b) 31F, if and only if 4|n.
(c) 4|F, if and only if 6 n.
(d) 5F, if and only if 5 n.
[Hint: All these require the aid of the previous
problem.]
8. Show that the sum of the squares of the first n
Fibonacci numbers is given by the formula
ing
F2+F+F. .+ F= F,F+
1
[Hint: Note that F F,(Fn+1 - F1-1)
F,F +1- FrFn-1.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc27b03fb-27ff-4129-a29b-052f3c3c0c2f%2F32295b36-63fd-4fbd-b186-93d5b59f681a%2Fbyy059p.jpeg&w=3840&q=75)
Transcribed Image Text:295
For any primep 2 or 5, it is known that either Fp-1
or Fp+1 is divisible by p. Confirm this in the case of
the primes 7, 11, 13, and 17.
3.
From the formula F+ F- F2 (-1)", conclude
that consecutive Fibonacci numbers are relatively
prime.
4.
n
One can prove that the greatest common divisor of two
Fibonacci numbers is also a Fibonacci number;
5.
specifically,
where d
gcd (n, m).
gcd (F1, Fm)= Fd
no
ner.
Verify this identity in the case of gcd(F9, F12) and
gcd(F15, F20).
mula
Use Problem 5 to prove that for n > 2, Fn| Fm if and
only if n m.
6.
no2
7. Establish each of the following assertions:
2|F, (that is, F, is even) if and only if 3 n.
(b) 31F, if and only if 4|n.
(c) 4|F, if and only if 6 n.
(d) 5F, if and only if 5 n.
[Hint: All these require the aid of the previous
problem.]
8. Show that the sum of the squares of the first n
Fibonacci numbers is given by the formula
ing
F2+F+F. .+ F= F,F+
1
[Hint: Note that F F,(Fn+1 - F1-1)
F,F +1- FrFn-1.]
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