Define $f:\mathbb N \to \mathbb N$ by
$f(n)=3n+1$ if $n$ is odd $f(n)=n/2$ if $n$ is even.Now can we show that for every $x$ in $\mathbb N$ there is $k$ in $\mathbb N$ such that $f^k(x)=1$ where $f^r=f \circ f^{r-1}$ and $f^1=f$.I could verify for some numbers like $6,3,10$ etc that this holds,but in general I could neither prove nor disprove it.I am new in these things,so I do not know much about how to solve such sequence problems.Can someone please provide me with a solution?
You have found the notorious Collatz conjecture which was proposed in 1937 and remains unsolved/open. The Wikipedia article (linked above) is good and has many pretty graphs etc.