Resolution of the $3n+1$ Problem Using Inequality Relation Between Indices of 2 and 3

Collatz conjecture states that an integer $n$ reduces to $1$ when certain simple operations are applied to it. Mathematically, it is written as $2^z = 3^kn + C$, where $z, k, C \geq 1$. Suppose the integer $n$ violates Collatz conjecture by reappearing, then the equation modifies to $2^z n =3^kn +C$. The article takes an elementary approach to this problem by stating that the inequality $2^z > 3^k$ must hold for $n$ to violate the Collatz conjecture. It leads to the inequality $z > 3k/2$ that helps obtain bounds on the value of $3^k/2^z$ and $2^z - 3^k . It is found that the $3n+1$ series loops for $1$ and negative integers. Finally, it is proved that the $3n+1$ series shows pseudo-divergence but eventually arrives at an integer less than the starting integer.