On the Glide of 3x+1 Problem

For any positive integer $n$, define an iterated function $$ f(n)=left{begin{array}{ll} n/2, & mbox{$n$ even,} 3n+1, & mbox{$n$ odd.} end{array} right. $$ Suppose $k$ (if it exists) is the lowest number such that $f^{k}(n)<n$, and there are $O(n)$ multiply by three and add one and $E(n)$ divide by two from $n$ to $f^{k}(n)$, then there must be $$ 2^{E(n)-1}<3^{O(n)}<2^{E(n)}. $$ Our results confirm the conjecture proposed by Terras in 1976.