Respuesta :
It is proved that every natural number can be expressed as the sum of distinct powers of 2 using strong induction.
What is natural number?
Natural numbers are those in mathematics that are utilized for counting and ordering. Cardinal numbers are those used for counting, while ordinal numbers are those used for ranking. The positive integers, also referred to as non-negative integers, are a subset of the natural numbers. A few examples are 1, 2, 3, 4, 5, 6,.... In other words, the set of all whole numbers other than 0 (i.e., 23, 56, 78, 999, 100202, etc.) is known as the natural numbers.
Here,
The statement is obviously true for n=0.
Assume that we are given an n≥1 and that it is true for all m with 0≤m<n.
When n=2^m then m<n and therefore m=∑k2^pk with finitely many pk, all of them different. It follows that n=∑2^(pk+1) with all pk+1 different.
When n=2^(m+1) with an m as before then n=2^0+∑k2^(pk+1) with all pk+1 different and different from 0.
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