How to prove distinct Fibonacci numbers

Question

8) How to prove that every positive integer can be written as the sum of one or more distinct non-consecutive Fibonacci numbers?

9) How to prove that the sum of any set of distinct non-consecutive Fibonacci numbers whose largest member is Fk is strictly less than Fk+1?

10) How to prove that no positive integer can be written in two different ways as the sum of distinct non-consecutive Fibonacci numbers?

11) How to show that the set of positive integers can be partitioned into Fibonacci sets (i.e. there's a collection of Fibonacci sets so that every positive integer is in precisely one of them) by referencing questions 8 and 10?

Answer 1

a.  since the beginning of the Fibonacci sequence is 0,1,1,2,3,5, etc. it's obvious that every positive integer can be expressed as the sum of ONE OR MORE distinct non-consecutive Fibonacci numbers. example: 4 = 1 + 3; 6 = 1+5; 7 = 2+5; 9 = 1+3+5

 

b. Let F(k+1) be 55; then the sum of these non-consecutive Fibonacci predecessors 34, 13, 5, 2, 1 is also 55. 34 is F(k), and this sum is NOT STRICTLY LESS THAN F(k+1). I believe that this counterexample disproves the second assertion in your submitted 'question.'