Proving if triangle with sides $a^n,b^n,c^n$ exists for all $n\in N$ then $a=b$ or $a=b=c$

Question

Given $a\ge b\ge c\ge0$ and number $n\in N$

If there exists a triangle with sides $a^n,b^n,c^n \,\,\,\forall n \in N$ then prove that $a=b$ or $a=b=c$ that is triangle is isosceles.

I vaguely have the idea that since $a^n$ follows exponential growth over $n$, $a^n$ grows faster than $b^n+c^n$ iff $a>b$

However, I dont know how to prove it rigoursly. I tried differentiating but couldn't produce an inequality.

Any hints appreciated.

Answer 1

Suppose $a\gt b\ge c$ and put $a=b+d$ with $d\gt 0$

Now for $n\ge 2$ we have $(b+d)^n\gt b^n+ndb^{n-1}$ and if $n\gt \frac cd$ then $$a^n=(b+d)^n\gt b^n+ndb^{n-1}\gt b^n+c^n$$

So we cannot have $a\gt b\ge c$ and we must have $a=b$