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If An=An-1/An-2 (n3), A1=1, and A2=-2, then A100=?

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If An=An-1/An-2 (n3), A1=1, and A2=-2, then A100=?

by Max@Math Revolution » Thu Jul 19, 2018 12:51 am

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[GMAT math practice question]

If An=An-1/An-2 (n3), A1=1, and A2=-2, then A100=?

A. 1
B. -1
C. 2
D. -2
E. -1/2
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by GMATGuruNY » Thu Jul 19, 2018 3:00 am
Max@Math Revolution wrote:[GMAT math practice question]

If An=An-1/An-2 (n3), A1=1, and A2=-2, then A100=?

A. 1
B. -1
C. 2
D. -2
E. -1/2
A� = 1
Aâ‚‚ = -2
A₃ = -2/1 = -2
Aâ‚„ = -2/-2 = 1
Aâ‚… = 1/-2 = -0.5
A₆ = -0.5/1 = -0.5
A₇ = -0.5/-0.5 = 1
A₈ = 1/-0.5 = -2

As illustrated by the results in blue, A� through A₆ represent one complete 6-term CYCLE:
1, -2, -2, 1, -0.5, -0.5
Starting with A₇ and A₈, the 6-term cycle begins again:
1, -2...

Since 6*16 = 96, the 96th term is the end of the 16th 6-term cycle.
Starting with A₉₇, the 6-term cycle begins again:
A₉₇ = 1
A₉₈ = -2
A₉₉ = -2
A�₀₀ = 1

The correct answer is A.
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by Max@Math Revolution » Sun Jul 22, 2018 5:05 pm
=>
A1 = 1, A2 =-2.
A3 = A2 / A1 = (-2)/1 = -2.
A4 = A3 / A2 = (-2)/(-2) = 1.
A5 = A4 / A3 = 1/(-2) = -(1/2).
A6 = A5 / A4 = -(1/2) / 1 = -(1/2).
A7 = A6 / A5 = -(1/2) / -(1/2) = 1 = A1.
A8 = A7 / A6 = 1 / (-1/2) = -2 = A2.
Thus, every 6th term is the same. That is,
A1 = A7 = A13 = ... = 1.
A2 = A8 = A14 = ... = -2.
A3 = A9 = A15 = ... = -2.
A4 = A10 = A16 = ... = 1.
A5 = A11 = A17 = ... = -(1/2).
A6 = A12 = A18 = ... = -(1/2).
Now, 100 = 6 * 16 + 4. Thus,
A100 = A4 = 1.
Therefore, the answer is A.
Answer: A
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