Fibonacci and the Pythagorean Problem 3. For any 6.3 Problems or F P+1 the prim 1. It can be established that each positive integer is representable as a sum of Fibonacci numbers, none taken more than once; for example, 4. From th that con prime. 5 = F3 + F4, 5. One car Fibonac 6 F1+ F3+ F4, specific 7=F1+ F2+ F3 + F4. gcd Write the integers 50, 75, 100, and 125 in this manner. Verify gcd(F1 Show that the sum of the first n Fibonacci (a) 2. numbers with odd indices is given by the formula 6. Use Pr F1+F3+ F5 + F2n-1 F2n only if CLST [Hint: Add the equalities F1 F2, F3 = F4- F2, Fs F6- Fa, ... .] (b) Show that the sum of the first n Fibonacci numbers with even indices is given by the 7. Establi 1 (a) 2 (b) 3 (c) (d) S formula 2U 1s F2+ F4+F6+ ..+F2n [Hint: = F2n+1 - 1. proble [Hint: Use part (a) and the identity FI+F2+ F3 + . + Fan = F2n+2 -- 1.} C Obtain the following formula for the altermaing um of Fibonacci numbers: 8 Show Fibon +(1)"+ Fi-F2 + F3- F4+ Wis = (-1)"+1 Fn-1 + 1.

Question

Number 1

Fibonacci and the Pythagorean Problem
3.
For any
6.3 Problems
or F
P+1
the prim
1. It can be established that each positive integer is
representable as a sum of Fibonacci numbers, none
taken more than once; for example,
4. From th
that con
prime.
5 = F3 + F4,
5.
One car
Fibonac
6
F1+ F3+ F4,
specific
7=F1+ F2+ F3 + F4.
gcd
Write the integers 50, 75, 100, and 125 in this manner.
Verify
gcd(F1
Show that the sum of the first n Fibonacci
(a)
2.
numbers with odd indices is given by the formula
6. Use Pr
F1+F3+ F5 + F2n-1 F2n
only if
CLST
[Hint: Add the equalities F1 F2,
F3 = F4- F2, Fs F6- Fa, ... .]
(b) Show that the sum of the first n Fibonacci
numbers with even indices is given by the
7. Establi
1
(a) 2
(b) 3
(c)
(d) S
formula
2U 1s
F2+ F4+F6+ ..+F2n
[Hint:
= F2n+1 - 1.
proble
[Hint: Use part (a) and the identity
FI+F2+ F3 + . + Fan = F2n+2 -- 1.}
C Obtain the following formula for the altermaing
um of Fibonacci numbers:
8 Show
Fibon
+(1)"+
Fi-F2 + F3- F4+
Wis
= (-1)"+1 Fn-1 + 1.

Image Transcription

Fibonacci and the Pythagorean Problem 3. For any 6.3 Problems or F P+1 the prim 1. It can be established that each positive integer is representable as a sum of Fibonacci numbers, none taken more than once; for example, 4. From th that con prime. 5 = F3 + F4, 5. One car Fibonac 6 F1+ F3+ F4, specific 7=F1+ F2+ F3 + F4. gcd Write the integers 50, 75, 100, and 125 in this manner. Verify gcd(F1 Show that the sum of the first n Fibonacci (a) 2. numbers with odd indices is given by the formula 6. Use Pr F1+F3+ F5 + F2n-1 F2n only if CLST [Hint: Add the equalities F1 F2, F3 = F4- F2, Fs F6- Fa, ... .] (b) Show that the sum of the first n Fibonacci numbers with even indices is given by the 7. Establi 1 (a) 2 (b) 3 (c) (d) S formula 2U 1s F2+ F4+F6+ ..+F2n [Hint: = F2n+1 - 1. proble [Hint: Use part (a) and the identity FI+F2+ F3 + . + Fan = F2n+2 -- 1.} C Obtain the following formula for the altermaing um of Fibonacci numbers: 8 Show Fibon +(1)"+ Fi-F2 + F3- F4+ Wis = (-1)"+1 Fn-1 + 1.

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