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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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Math Expert
Joined: 02 Sep 2009
Posts: 58401
Let a_1, a_2, ... be a sequence determined by the rule a_n=a_{n-1}/2 i  [#permalink]

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27 Mar 2019, 00:29
00:00

Difficulty:

95% (hard)

Question Stats:

25% (03:12) correct 75% (02:21) wrong based on 24 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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Math Expert
Joined: 02 Aug 2009
Posts: 7969
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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27 Mar 2019, 01:34
1
1
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

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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