Collatz-ish
A sequence is generated using these rules. For which values of n is the nth term equal to n?
Age
14 to 16
Challenge level
Problem
The first term of a sequence of positive integers is $6$. The other terms in the sequence follow these rules:
if a term is even then divide it by $2$ to obtain the next term;
if a term is odd then multiply it by $5$ and subtract $1$ to obtain the next term.
For which values of $n$ is the $n$th term equal to $n$?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
The terms of the sequence are$$6, 3, 14, 7, 34, 17, 84, 42, 21, 104, 52, 26, 13, 64, 32, 16, 8, 4, 2, 1, 4, 2, 1, \dots$$As can be seen, there will now be no other terms in the sequence other than $4$, $2$, and $1$. It can also be seen that the only values of $n$ for which the $n$th term equals $n$ are $13$ and $16$.