sachin123 said:Spec of root 2 is that set of elements floor(k * (root 2)) ; k >= 0 .
I have no idea of how I can prove the statement in the question.
Prove that spec of root 2 contains infinitely many powers of 2.
I need ideas on how to proceed.
Thank you
sachin123 said:Here, it means Spec√2= {⌊k√2⌋ : k≥0}
Dick said:Yeah, I'm actually running into the same problem. It's pretty easy to show 2^n*sqrt(2)-floor(2^n*sqrt(2)) is less than 1/2 an infinite number of times and I thought that would cover it, but I keep getting an inequality pointing the wrong way.
The "spec of root 2" refers to the set of all numbers that can be obtained by taking powers of 2 and multiplying them by the square root of 2.
We can prove this by showing that for every natural number n, there exists a power of 2 that is a member of the set. This can be done through mathematical induction, where we show that for n=1, there exists a power of 2 in the set, and then assume that for some arbitrary n=k, there exists a power of 2 in the set, then show that for n=k+1, there also exists a power of 2 in the set.
Yes, there is a formula that can be used to prove this statement. It is called the geometric series formula, which states that for a geometric series with starting term a and common ratio r, the sum of n terms is given by a(1-r^n)/(1-r). We can use this formula to show that for every natural number n, there exists a power of 2 in the set.
Yes, there are other mathematical concepts and techniques that can be used to prove this statement. For example, we can use the concept of limit to show that as n approaches infinity, the sum of n terms in the geometric series formula becomes infinitely large, indicating that there are infinitely many powers of 2 in the set.
Proving this statement is important because it helps us understand the behavior of numbers and their relationships to one another. It also allows us to make connections between different mathematical concepts, such as powers and limits, and helps us build a stronger foundation for future mathematical discoveries and advancements.