In an exercise, I had to prove by induction that $(x^n-y^n)$ is divisible by $(x-y)$ for all $n\in\mathbb{N}$. But I don't understand one of the algebraic manipulation, and to give you more context, here's the given solution:
Solution:
Let the given statement be P(n). Then,
P(n): $(x^n - y^n)$ is divisible by $(x - y)$.
When $n = 1$, the given statement becomes: $(x^1 - y^1)$ is divisible by $(x-y)$, which is clearly true.
Therefore P(1) is true.
Let p(k) be true. Then,
P(k): $x^k - y^k$ is divisible by $(x-y)$.
Now, $x^{k + 1}$ - $y^{k + 1}$ = $x^{k + 1}$ - $x^ky$ - $y^{k + 1}$
[on adding and subtracting x)ky]= ${x^k}(x - y) + y(x^k - y^k)$, which is divisible by (x - y) [using (i)]
⇒ P$(k + 1)$: $x^{k + 1} - y^{k + 1}$ is divisible by $(x-y)$
⇒ P$(k + 1)$ is true, whenever P(k) is true.
Thus, P(1) is true and P$(k + 1)$ is true, whenever P(k) is true.
Hence, by the Principal of Mathematical Induction, P(n) is true for all n $\in$ $\mathbb{N}$
I don't understand why $x^{k + 1} - y^{k + 1} = x^{k + 1} - x^ky - y^{k + 1}$, like why is it that we can transform it like that, because where does "$- x^ky$" come from?
I'm also very sorry if it's unclear, so here is the link for the exercise: https://www.math-only-math.com/proof-by-mathematical-induction.html
And it should be the 5th example. Thank you so much!
It is a typo.
The following equality is clearly false unless $x=0$ or $y=0$: $$x^{k + 1} - y^{k + 1} = x^{k + 1} - x^ky - y^{k + 1}.$$ Not only is that equality blatantly false, but it's not consistent with what follows in the proof.
The intended equality was: $$x^{k + 1} - y^{k + 1} = x^{k + 1} - x^ky + x^ky - y^{k + 1}.$$