let x=sum(10^k,k=0 to n-1)=(10^n-1)/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^n-1)/9 and simplifying we get on the left side
(10^n+2)(10^n-1)/9
(10^(2n)+10^n-2)/9
on the right side
9*(10^n-1)^2/81 + 3*(10^n-1)/9
[(10^n-1)^2+3*(10^n-1)]/9
[10^(2n)-2*10^n+1+3*10^n-3)/9
(10^(2n)+10^n-2)/9
thus the left and right hand sides simplify to the same equation and thus it holds for all n
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Posted by Daniel
on 2013-09-05 11:19:23 |
solution