Inspired by Satyajit Mohanty

Algebra Level 5

\[{ 3 }^{ x }+{ 3 }^{ x+1 }+\cdots+{ 3 }^{ x+31032001 }={ 27 }^{ x }+{ 27 }^{ x+1 }+\cdots+{ 27 }^{ x+31032001 }\]

If the above equation is true for some integer \(x\) and it can be expressed in the form of:-

\[\large{\frac { \log _{ A }{ \left( \frac { B }{ { A }^{ C }+{ A }^{ D }+1 } \right) } }{ 2 }} \]

where \(A,B,C \) and \(D\) are positive integers, find \(A+B+C+D\).


Inspiration.

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