Using the principle of mathematical induction, prove that x^n - y^n is divisible by (x-y) for all n∈N.

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> [!TLDR]
> Base case $n=1$ is trivial; the inductive step factors $x^{k+1}-y^{k+1}$ as $(x-y)$ times an integer combination of the inductive-hypothesis quotient.
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> Let $P(n)$: '$x^n-y^n$ is divisible by $(x-y)$'.
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> **Base case ($n=1$):** $x^1-y^1=x-y$, which is divisible by $(x-y)$ (quotient $1$). So $P(1)$ is true.
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> **Inductive hypothesis:** assume $P(k)$ is true for some $k\in\mathbb N$, i.e. $x^k-y^k=(x-y)\cdot m$ for some expression $m$.
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> **Inductive step:** show $P(k+1)$ holds.
> $$x^{k+1}-y^{k+1}=x^{k+1}-xy^k+xy^k-y^{k+1}=x(x^k-y^k)+y^k(x-y)$$ …

Q.Using the principle of mathematical induction, prove that x^n - y^n is divisible by (x-y) for all n∈N.

Concept understanding — Proving Divisibility by Induction

Q.Using the principle of mathematical induction, prove that x^n - y^n is divisible by (x-y) for all n∈N.

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