While playing with numbers, I thought of type of numbers, and then the first thing came into mind was $\text{Odd}$ and $\text{Even}$.
I observed a very interesting fact that any $x\in\Bbb{N}$ can be converted into $1$ by applying these two operations.
$\text{If the number is odd:}$ $$\text{Multiply it by $3$ and add 1}\tag{$3k+1$}$$
$\text{If the number is even:}$ $$\text{Divide it by 2}\tag{$\frac{k}{2}$}$$
For Example: $$6\tag{Even}$$ $$\frac62=3\tag{Odd}$$ $$3\times3+1=10\tag{Even}$$ $$\frac{10}2=5\tag{Odd}$$ $$3\times5+1=16\tag{Even}$$ $$\frac{16}2=8\tag{Even}$$ $$\frac{8}2=4\tag{Even}$$ $$\frac{4}2=2\tag{Even}$$ $$\frac{2}2=1$$ I checked it $2-30$ and found my observation true. Then by taking a general case by letting the number $k$ but can't reach on any result since it form a infinite series of operations.
Please help me prove this or help me disprove it by giving a counter example!!!
This is a well-known and widely studied problem. But essentially nothing is known. Lagarias has a well-known annotated bibliography on various considerations of the problem, but the theme is this --- we don't know whether or not every sequence reaches 1, nor do we have a good idea as to how to determine whether this is or isn't true.