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- Feb 14, 2012

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I am unable to determine the value of \(\displaystyle a_{1000}\) in the problem as stated below because I think I failed to observe another useful pattern of the given sequence.

Could anyone please help me out with this problem? Thanks in advance.

Problem:

A sequence \(\displaystyle a_1\), \(\displaystyle a_2\), \(\displaystyle a_3,\;\cdots\) of positive integers satisfies the following properties:

\(\displaystyle a_1=1\)

\(\displaystyle a_{3n+1}=2a_n+1\)

\(\displaystyle a_{n+1}\ge a_n\)

\(\displaystyle a_{2001}=200\)

Find the value of \(\displaystyle a_{1000}\)