Prove the following Identities(Discrete math Fibonnacci Sequence Question)

Question

First question:

F₀²+F₁²+F₂²+...+F_n²=F_n*F_n+1

Second question:

F_n-1*F_n+1-F_n²=(-1)ⁿ

Third Question

Choose an integer uniformly from the set {1,2,3,...,30}. Let A be the event that it is divisible by 2; let B be the event that it is divisible by 3; let C be the event that it is divisible by 7.

Part A) determine the probabilities of A,B,C

Part B) Which of the pairs (A,B),(B,C), and(A,C) are independent?

Answer 1

proof by mathematical induction

0,1,1,2,3,5.8.13,21,34,...

1+3+9=14

F1=0

0+1=1=F1

1+1=2=F2

1+2=3=F3

sum 1st 3 squared 14=5(3)-1=F3F4-1=15-1

2+3=5=F4

3+5=8

sum 1st 4 squared = 39 =F5F4-1= 40-1

assume true for any Fk-1

then show true for Fk

1+2+3+5+...+Fk-2 +Fk-1 = Fk-2Fk-1 -1

show

1+2+3+5...+Fk-2 + Fk-1 +Fk = Fk-1Fk -1

=Fk-2Fk-1 -1 + Fk=Fk-1(Fk-2) -1

Fk-2Fk-1 = Fk-2Fk-1

QED