a specific sum of cube property(narcissistic number)

Question

So I am currently doing a problem about narcissistic number and I need to prove that $a^3+b^3+c^3=a*10^{2n}+b*10^n+c$, where $a=1666..666, b=5000..00, c=33...333 $ and $a,b,c$ has $n$ digits. I am trying to do an induction and have observed that $3a+2=b$, $3c+1=2b$ and $a+b+c=10^n$; the difficult part when doing induction is I just cannot write the equation into $b$ and $n$ which would allow me to solve as I know that $b=(10^n)/2$. Is there any idea that is better than induction? Thanks in advance.