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# A sequence is given by the rule a(n) = |a_{(n−2)}| – |a_{(n−1)}| for a

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Math Expert
Joined: 02 Sep 2009
Posts: 52390
A sequence is given by the rule a(n) = |a_{(n−2)}| – |a_{(n−1)}| for a  [#permalink]

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07 Jul 2018, 08:39
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Difficulty:

45% (medium)

Question Stats:

64% (02:17) correct 36% (01:55) wrong based on 36 sessions

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A sequence is given by the rule $$a_n = |a_{(n−2)}| – |a_{(n−1)}|$$ for all $$n\geq 3$$, where $$a_1 = 0$$ and $$a_2 = 3$$. What is the value of $$a_{99}$$?

A. -297
B. -3
C. 0
D. 3
E. 297

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Math Expert
Joined: 02 Aug 2009
Posts: 7212
A sequence is given by the rule a(n) = |a_{(n−2)}| – |a_{(n−1)}| for a  [#permalink]

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07 Jul 2018, 09:11
Bunuel wrote:
A sequence is given by the rule $$a_n = |a_{(n−2)}| – |a_{(n−1)}|$$ for all $$n\geq 3$$, where $$a_1 = 0$$ and $$a_2 = 3$$. What is the value of $$a_{99}$$?

A. -297
B. -3
C. 0
D. 3
E. 297

GMAT does not expect us to calculate too much so there has to be a sequence...

$$a_n = |a_{(n−2)}| – |a_{(n−1)}|$$ ........$$a_3 = |a_{(3−2)}| – |a_{(3−1)}|.............a_3=|a_1|-|a_2|..........a_3=0-3=-3$$ ...
$$a_4 = |a_2| – |a_3|=|3|-|-3|=0$$ ..........
$$a_5 = |a_3| – |a_4|=|-3|-|0|=3$$ ..........
so the series is 0,3,-3,0,3,-3,.......
we are looking for 99 and the cyclicity happens every 3 numbers so $$a_{99}=a_3=-3$$

B
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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Re: A sequence is given by the rule a(n) = |a_{(n−2)}| – |a_{(n−1)}| for a  [#permalink]

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07 Jul 2018, 09:23
Given :
an=|a(n−2)|–|a(n−1)|an=|a(n−2)|–|a(n−1)| for all n≥3n≥3
where, a1=0 and a2=3
a99 = ?

Solving the equation we get:
a3 = -3
a4 = 0
a5 = 3
a6 = -3

See the repeated pattern yet ?
a1, a2, a3, a4, a5, a6 = 0, 3, -3, 0, 3, -3

Since, a99 is the multiple of 3, so a99 should be the third term in the pattern, in this case its -3.

Hence, B.
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Re: A sequence is given by the rule a(n) = |a_{(n−2)}| – |a_{(n−1)}| for a &nbs [#permalink] 07 Jul 2018, 09:23
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