vela said:That's not right. If you let ##n \rightarrow n+1##, you get
\begin{align*}
x^{n+1}-y^{n+1} &= (x-y)(x^{(n+1)-1}+x^{(n+1)-2}y+\cdots+xy^{(n+1)-2}+y^{(n+1)-1}) \\
&= (x-y)(x^n+x^{n-1}y+\cdots+xy^{n-1}+y^{n})
\end{align*} You tacked on the last two terms, but all of the other terms in the sum don't match.
An induction proof statement is a mathematical proof that uses the principle of mathematical induction to establish the truth of a statement for all natural numbers. It involves showing that a statement holds for a base case (usually n = 1) and then using the inductive hypothesis to show that it also holds for n+1.
The purpose of using induction in a proof is to establish the truth of a statement for all natural numbers, without having to explicitly prove it for each individual case. It allows for a more concise and efficient way of proving mathematical statements.
To prove this statement using mathematical induction, you would first show that it holds for the base case (n = 1). Then, assuming it holds for n, you would use the inductive hypothesis to show that it also holds for n+1. This would involve expanding the expression for (n+1) and simplifying it to match the original statement.
No, induction can only be used to prove statements that are true for all natural numbers. It cannot be used to prove statements that are only true for a finite number of cases, or for real numbers.
One common mistake to avoid when using induction in a proof is assuming that the statement is true for all natural numbers without properly proving it for the base case and using the inductive hypothesis to show it holds for all subsequent cases. It is also important to make sure that the inductive step is correct and that the statement holds for n+1, not just for n.