let x=sum(10^k,k=0 to n1)=(10^n1)/9
then the left hand side is
10^n*x+2x=(10^n+2)x
the right hand side is
(3x+1)*(3x)=9x^2+3x
where n is the number of 1's appearing on the left side
Now substituting in the equation for x=(10^n1)/9 and simplifying we get on the left side
(10^n+2)(10^n1)/9
(10^(2n)+10^n2)/9
on the right side
9*(10^n1)^2/81 + 3*(10^n1)/9
[(10^n1)^2+3*(10^n1)]/9
[10^(2n)2*10^n+1+3*10^n3)/9
(10^(2n)+10^n2)/9
thus the left and right hand sides simplify to the same equation and thus it holds for all n

Posted by Daniel
on 20130905 11:19:23 