<p>Let <img alt="$a_1,a_2,\ldots$" height="11" src="https://latex.artofproblemsolving.com/4/3/e/43ec2086760ce93aaca1938b12c93cbf7cf9dc76.png" width="70" /> be a sequence determined by the rule <img alt="$a_n=a_{n-1}/2$" height="18" src="https://latex.artofproblemsolving.com/7/b/1/7b1c2e08f6bbbcf7c5cc64f53365cdbb19c9e1b1.png" width="96" /> if <img alt="$a_{n-1}$" height="10" src="https://latex.artofproblemsolving.com/8/9/6/896d56965b17443a55b2b5e55f798adf6465380d.png" width="33" /> is even and <img alt="$a_n=3a_{n-1}+1$" height="15" src="https://latex.artofproblemsolving.com/d/8/d/d8df54e3b84063c1be0bfae4f7daf50d48a235c2.png" width="118" /> if <img alt="$a_{n-1}$" height="10" src="https://latex.artofproblemsolving.com/8/9/6/896d56965b17443a55b2b5e55f798adf6465380d.png" width="33" /> is odd. For how many positive integers <img alt="$a_1 \le 2008$" height="15" src="https://latex.artofproblemsolving.com/b/0/3/b03a47d044f9e0bd9666519d7c79affb9f00f5fb.png" width="77" /> is it true that <img alt="$a_1$" height="10" src="https://latex.artofproblemsolving.com/1/a/b/1ab39d761413804680d26d972381f028001562f5.png" width="15" /> is less than each of <img alt="$a_2$" height="10" src="https://latex.artofproblemsolving.com/5/e/9/5e97a8af68fbc8e357d3ee0eba452022b06c1875.png" width="15" />, <img alt="$a_3$" height="10" src="https://latex.artofproblemsolving.com/5/7/2/572b65ebc3f438b176b4ca4a890799f2f72564af.png" width="15" />, and <img alt="$a_4$" height="10" src="https://latex.artofproblemsolving.com/5/e/2/5e28850813a5321d2a8ae3c0510457a7337975ed.png" width="16" />?</p>

Question No: 249

Let $a_1,a_2,\ldots$ be a sequence determined by the rule $a_n=a_{n-1}/2$ if $a_{n-1}$ is even and $a_n=3a_{n-1}+1$ if $a_{n-1}$ is odd. For how many positive integers $a_1 \le 2008$ is it true that $a_1$ is less than each of $a_2$ , $a_3$ , and $a_4$ ?

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Tags: Algebra

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