Question

2. Some facts about Fibonacci sequence: 0,1,1,2,3,5, 8, 13,21,34,55, 89, for n 0 for n 1 F-1 Ffor n22 what is the lest value of n for which F, > 100? what is the least alle urn ir which F > 10002 Let An (F+F2+..Fl/n be the average of the first n Fibonacci numbers. What is the least value of n for which An 10? Find all n for which F, = n, Explain why these are the only cases. Find all n for which Fn = na Explain why these are the only cases. 3. Prove the following identity for n 22: F1 Fa- Fa+ F-2 Prove the following identity for n 2 1: Fa+1 F2+12 4. Define the following (almost Fibonacci) recurrence for n =0 for n 1 G-1+ Gn-2+1 for n 22 G1 Find the values of Go,Gi,-., G10- Express G as a function of Fibonacci numbers. Prove that your expression for Gn is correct for all n 2 0. 5. Find at least three interesting and fascinating facts or properties about the Fibonacci bers and/or the Golden Ratio that do not appear in the class presentation. You do not need to provide proofs. Support your findings with pointers to their resources.

Answer 1

2) The Fibonacci numbers is given by

0112358314594370774 123584371898 1236955 2 012345678

Therefore the least value of n for which F_{​​​​​​n} > 100 is 12 and that of F_{​​​​​​n} > 1000 is 17.

We have $A_1 = 1; A_2 = 1; A_3 = \frac{4}{3} ; A_4 = \frac{7}{4} ; A_5 = \frac{12}{5}; A_6 = \frac{20}{6} ; A_7 = \frac{33}{7}$

$A_8 = \frac{54}{8} ; A_9 = \frac{88}{9} ; A_{10} = \frac{143}{10} > 10$ . Therefore least value of n for which A_{​​​​​​n} > 10 is 10.

For n = 0, 1, 5 we have F_{​​​​​​n} = n and for 1 < n < 5 we have F_{​​​​​​n} < n and for all n > 5, F_{​​​​​​n} > n, hence these are the only cases in which F_{​​​​​​n} = n.

For n = 0, 1, 12 we have F_{​​​​​​n} = n² and for 1 < n < 12 we have F_{​​​​​​n} < n² and for n > 12 we have n² < F_{​​​​​​n} , therefore these are the only cases in which F_{​​​​​​n} = n².