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Prove the following is valid only when n is an odd integer.
x^n + 1= (x+1)(x^n - X^n-2 + ... + (x^2 - x + 1).
It's an easy 3 line proof.
x^n + 1= (x+1)(x^n - X^n-2 + ... + (x^2 - x + 1).
It's an easy 3 line proof.
mathwonk said:robert, how can you tell he meant the converse of what he said? what he said was to prove his statement false for even n, not to prove it true for odd n. that at least is how i translate the word "only".
The proof for x^n + 1 being valid for odd n is based on mathematical induction. It involves showing that the statement is true for n=1, and then assuming it is true for some arbitrary odd number k. By using algebraic manipulation and the properties of odd numbers, it can be shown that the statement is also true for k+2. This completes the proof by induction.
Mathematical induction is necessary for this proof because it allows for a general statement to be proven true for an infinite number of cases. In this case, we are proving that x^n + 1 is valid for all odd values of n, which is an infinite number of cases. Mathematical induction provides a systematic way to prove this statement for all possible values of n.
No, this proof cannot be extended to even values of n. This is because the statement x^n + 1 is only valid for odd values of n. When n is an even number, the statement becomes x^n + 1 = x^even + 1 = (x^2)^k + 1, which is not true for all values of x. Therefore, this proof is only valid for odd values of n.
This proof has several practical applications in mathematics and computer science. It is used in fields such as number theory, algebra, and cryptography. It is also used in computer algorithms that involve exponents and finite fields. Additionally, this proof can be used to simplify complex expressions and solve equations involving odd exponents.
Yes, this proof can be applied to other similar statements involving odd exponents. For example, it can be extended to the statement x^n - 1 is valid for odd n, or x^n + 2 is valid for odd n. The same principles of mathematical induction can be used to prove these statements for all odd values of n.