Proving the Sum of Odd Numbers in Number Theory Problem | Homework Statement

In summary, the problem is to show that for every odd positive integer n, the equation xn + yn = (x+y)(xn-1 - xn-2y + xn-3y2 - ... - xyn-2 + yn-1) holds true. The suggested approach is to expand the right hand side and compare it to the left hand side, rather than using induction.
  • #1
Abst.nonsense
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Homework Statement



Show that for every odd positive integer n the following is correct

xn + yn = (x+y)(xn-1 - xn-2y + xn-3y2 - ... - xyn-2 + yn-1)

Homework Equations



The one above.

The Attempt at a Solution



I have an idea about using induction to prove this. My idea is to write the RHS as a sum where k goes from 1 to n, and also write a similar sum where k goes from 1 to (n+2) (the next odd number). But it really have stopped there.

Since my experience with mathematical proofs is pretty nonexistent, I would appreciate any help on how to attack this problem.
 
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  • #2
No need to use induction, just go ahead and multiply out the right hand side.
 

What is the problem statement?

The problem statement is to prove that the sum of the first n odd numbers is equal to n^2.

Why is this problem important in number theory?

This problem is important because it helps us understand the properties of odd numbers and their relationships to other numbers, as well as providing a foundation for more complex number theory problems.

What is the formula for the sum of odd numbers?

The formula for the sum of the first n odd numbers is n^2, where n represents the number of odd numbers being added.

What is an example of how to prove this statement?

One example of proving this statement is by using mathematical induction, where we show that the statement is true for a base case (n=1) and then assume it is true for n=k and prove it is also true for n=k+1.

What are some real-world applications of this concept?

This concept can be applied in various fields such as physics, computer science, and cryptography, where odd numbers play a significant role in calculations and algorithms.

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