Prove that $ a^n + b^n + c^n = d^n + e^n + f^n $ by induction If $a,b,c,d,e,f$ are six real numbers such that:
$$ a + b + c = d + e + f $$
$$ a^2 + b^2 + c^2 = d^2 + e^2 + f^2 $$
$$ a^3 + b^3 + c^3 = d^3 + e^3 + f^3 $$
Prove by mathematical induction that:
$$ a^n + b^n + c^n = d^n + e^n + f^n $$

I tried solving this question by correlating to
$$ a^k + b^k = (a + b)(a^{k-1} + b^{k-1}) - ab(a^{k-2} + b^{k-2}) $$
The problem is that terms with $abc$ do not come in common while expanding. Could you please help me solve this question?
This question originates from the level III excercise of SK Goyal's book of Algebra for JEE Mains and Advanced.

Consider the polynomial $p(x)=x^3-sx^2+ux-v$ where $s=a+b+c=d+e+f$, $u=ab+bc+ca=\frac 12\left((a+b+c)^2-a^2+b^2+c^2\right)=de+ef+fd$ and $v=abc=\frac 13\left((a^3+b^3+c^3)-s(a^2+b^2+c^2)+u(a+b+c)\right)=def$
Then $p(a)=p(b)=p(c)=p(d)=p(e)=p(f)=0$ and you can use $$a^rp(a)+b^rp(b)+c^rp(c)=0$$ to obtain an expression for the sum $a^{r+3}+b^{r+3}+c^{r+3}$ in terms of sums of lower powers and the common constants $s,u,v$. This can be used for the inductive step.
