Collatz 13
If a number is even, halve it; if odd, treble it and add 1. If a sequence starts at 13, what will be the value of the 2008th term?
Age
14 to 16
Challenge level
Problem
A sequence of positive integers $t_{1},t_{2}, t_{3}, t_{4}, ...$ is defined by:
$t_{1}=13$
$t_{n+1}=\frac{1}{2}t_{n}$ if $t_{n}$ is even
$t_{n+1}=3t_{n}+1$ if $t_{n}$ is odd.
What is the value of $t_{2008}$?
Student Solutions
The sequence proceeds as follows:
$t_{1} = 13$
$t_{2} = 40$
$t_{3} = 20$
$t_{4} = 10$
$t_{5} = 5$
$t_{6} = 16$
$t_{7} = 8$
$t_{8} = 4$
$t_{9} = 2$
$t_{10} = 1$
$t_{11} = 4$
$t_{12} = 2$
$t_{13} = 1$
The block $2, 1, 4$ repeats ad infinitum after $t_{8}$.
When $n$ is a multiple of $3$, $t_{n} = 2$.
Since $2007$ is a multiple of $3$, $t_{2008} = 1$