Final answer:
The proof uses induction to show that any positive integer can be expressed as a sum of distinct Fibonacci numbers without consecutive indices, according to Zeckendorf's Theorem.
Step-by-step explanation:
The question asks us to prove using induction that any positive integer can be expressed as a sum of distinct Fibonacci numbers such that no two of them are consecutive Fibonacci indices. This is a mathematical conjecture known as Zeckendorf's Theorem. To prove this by induction, we would start by showing it works for a base case, such as the number 1, which is a Fibonacci number itself (F1 or F2). Then, we assume it works for all integers up to an arbitrary positive integer k. Lastly, for the induction step, we must show that k+1 can also be expressed as such a sum, which involves finding the largest Fibonacci number less than or equal to k+1, subtracting it from k+1, and using the inductive hypothesis to express the remainder as the sum of distinct non-consecutive Fibonacci numbers.