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Let a_1, a_2, ... be a sequence determined by the rule a_n=a_{n-1}/2 i

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Let a_1, a_2, ... be a sequence determined by the rule a_n=a_{n-1}/2 i  [#permalink]

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New post 27 Mar 2019, 00:29
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A
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Difficulty:

  65% (hard)

Question Stats:

20% (02:15) correct 80% (01:58) wrong based on 15 sessions

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

(A) 250
(B) 251
(C) 501
(D) 502
(E) 1004

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Re: Let a_1, a_2, ... be a sequence determined by the rule a_n=a_{n-1}/2 i  [#permalink]

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New post 27 Mar 2019, 01:34
Bunuel wrote:
Let \(a_1\), \(a_2\), ... be a sequence determined by the rule \(a_n=\frac{a_{n-1}}{2}\) if \(a_{n-1}\) is even and \(a_n=3a_{n-1}+1\) if \(a_{n-1}\) is odd. For how many positive integers \(a_1 \leq 2008\) is it true that \(a_1\) is less than each of \(a_2\), \(a_3\), and \(a_4\)?

(A) 250
(B) 251
(C) 501
(D) 502
(E) 1004


1) \(a_1\) cannot be EVEN, otherwise it will be > \(a_2\)..
So all even 2008/2=1004 gone.
2) Now , \(a_2\) cannot result in a multiple of 4, otherwise \(a_3\) will become less than \(a_1\).
For this 3a+1 shouldn't be multiple of 4. So a should not leave a remainder 1 when divided by 4. That means a cannot be 1, 5, 9 and so on.

Thus half the odd numbers possible ..3,7..etc
Answer 1004/2=502

D
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Re: Let a_1, a_2, ... be a sequence determined by the rule a_n=a_{n-1}/2 i   [#permalink] 27 Mar 2019, 01:34
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