9433
/www/nrich/html/content/id/9433/
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2012-09-18T12:31:11
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A sequence of positive integers $t_{1},t_{2}, t_{3}, t_{4}, ...$ is defined by:<br></br>
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$t_{1}=13$<br></br>
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$t_{n+1}=\frac{1}{2}t_{n}$ if $t_{n}$ is even<br></br>
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$t_{n+1}=3t_{n}+1$ if $t_{n}$ is odd.<br></br>
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What is the value of $t_{2008}$?<br></br>
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If you liked this problem, <a href="/6401">here is an NRICH task</a> which challenges you to use similar mathematical ideas.<br></br>
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The sequence proceeds as follows:<br></br>
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$t_{1} = 13$<br></br>
$t_{2} = 40$<br></br>
$t_{3} = 20$<br></br>
$t_{4} = 10$<br></br>
$t_{5} = 5$<br></br>
$t_{6} = 16$<br></br>
$t_{7} = 8$<br></br>
$t_{8} = 4$<br></br>
$t_{9} = 2$<br></br>
$t_{10} = 1$<br></br>
$t_{11} = 4$<br></br>
$t_{12} = 2$<br></br>
$t_{13} = 1$<br></br>
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The block $2, 1, 4$ repeats ad infinitum after $t_{8}$.<br></br>
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When n is a multiple of $3$, $t_{n} = 2$.<br></br>
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Since $2007$ is a multiple of $3$, $t_{2008} = 1$<br></br>
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<mdoxml version="1.0">Patterns and Sequences - Stage 4 Short Problem, UKMT 2008-2009 p118 Q15</mdoxml>
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Weekly Problem 46 - 2012
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?
Secondary Mapping Document
DisplayCabinet
Secondary Mapping Document
Patterns and sequences US