# Fibonacci and the Pythagorean Problem3.For any6.3 Problemsor FP+1the prim1. It can be established that each positive integer isrepresentable as a sum of Fibonacci numbers, nonetaken more than once; for example,4. From ththat conprime.5 = F3 + F4,5.One carFibonac6F1+ F3+ F4,specific7=F1+ F2+ F3 + F4.gcdWrite the integers 50, 75, 100, and 125 in this manner.Verifygcd(F1Show that the sum of the first n Fibonacci(a)2.numbers with odd indices is given by the formula6. Use PrF1+F3+ F5 + F2n-1 F2nonly ifCLST[Hint: Add the equalities F1 F2,F3 = F4- F2, Fs F6- Fa, ... .](b) Show that the sum of the first n Fibonaccinumbers with even indices is given by the7. Establi1(a) 2(b) 3(c)(d) Sformula2U 1sF2+ F4+F6+ ..+F2n[Hint:= F2n+1 - 1.proble[Hint: Use part (a) and the identityFI+F2+ F3 + . + Fan = F2n+2 -- 1.}C Obtain the following formula for the altermaingum of Fibonacci numbers:8 ShowFibon+(1)"+Fi-F2 + F3- F4+Wis= (-1)"+1 Fn-1 + 1.

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Number 1 help_outlineImage TranscriptioncloseFibonacci 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. fullscreen
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