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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 7603
GMAT 1: 760 Q51 V42 GPA: 3.82
If An=An-1/An-2, A1=1, and A2=-2, then A100=?  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 63% (02:25) correct 37% (02:40) wrong based on 56 sessions

### HideShow timer Statistics [GMAT math practice question]

If $$A_n = \frac{A_{n-1}}{A_{n-2}}$$, $$A_1=1$$, and $$A_2 =-2$$, then $$A_{100}=$$?

$$A. 1$$

$$B. -1$$

$$C. 2$$

$$D. -2$$

$$E. \frac{-1}{2}$$

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If An=An-1/An-2, A1=1, and A2=-2, then A100=?  [#permalink]

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5
Given A1=1 , A2= -2
then acc. to question we can find :
A3 = -2/1 = -2
A4= -2/-2 = 1
like wise :
A5 = -1/2
A6 = -1/2
A7= 1
A8= -2
.
.
.
now if you see after A6 cycle has started again , so for 100th term = 6*16 +4
cycle will repeat 16 times and A100 = A4

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Last edited by GMATbuster92 on 21 Jul 2018, 04:09, edited 2 times in total.
##### General Discussion
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Re: If An=An-1/An-2, A1=1, and A2=-2, then A100=?  [#permalink]

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1
MathRevolution wrote:
[GMAT math practice question]

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

$$A. 1$$
$$B. -1$$
$$C. 2$$
$$D. -2$$
$$E. \frac{-1}{2}$$

The question is not clear, MathRevolution - What does n3 mean in the question stem?
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Re: If An=An-1/An-2, A1=1, and A2=-2, then A100=?  [#permalink]

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pushpitkc wrote:
MathRevolution wrote:
[GMAT math practice question]

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

$$A. 1$$
$$B. -1$$
$$C. 2$$
$$D. -2$$
$$E. \frac{-1}{2}$$

The question is not clear, MathRevolution - What does n3 mean in the question stem?

Edited the question. I think now it reads as it should. Thank you.
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Re: If An=An-1/An-2 (n3), A1=1, and A2=-2, then A100=?  [#permalink]

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Quote:

If An=An-1/An-2 for all values n≥3, A1=1, and A2=-2, then A100=?

$$A. 1$$
$$B. -1$$
$$C. 2$$
$$D. -2$$
$$E. \frac{-1}{2}$$

$$A₁ = 1$$
$$A₂ = -2$$
$$A₃ = \frac{-2}{1} = -2$$
$$A₄ = \frac{-2}{-2} = 1$$
$$A₅ = \frac{1}{-2} = -0.5$$
$$A₆ = \frac{-0.5}{1} = -0.5$$

$$A₇ = \frac{-0.5}{-0.5} = 1$$
$$A₈ = \frac{1}{-0.5} = -2$$

As illustrated by the results in blue, A₁ 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 16*6 = 96, the 96th term represents the end of the 16th 6-term cycle.
Starting with A₉₇, the 6-term cycle begins again:
A₉₇ = 1
A₉₈ = -2
A₉₉ = -2
A₁₀₀ = 1

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Re: If An=An-1/An-2, A1=1, and A2=-2, then A100=?  [#permalink]

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Solution

Given:

• $$A_n = \frac{A_{n-1}}{A_{n-2}}$$
• $$A_1= 1$$
• $$A_2= -2$$

To find:
• $$A_{100}$$.

Approach and Working:

• $$A_3 =\frac{A_2}{A_1}= \frac{-2}{1}= -2$$
• $$A_4= \frac{A_3}{A_2}= \frac{-2}{-2}= 1$$
• $$A_5= \frac{A_4}{A_3}= \frac{1}{-2}= -0.5$$
• $$A_6= \frac{A_5}{A_4}= \frac{-0.5}{1}= -0.5$$
• $$A_7=\frac{A_6}{A_5}= \frac{-0.5}{-0.5}= 1$$
• $$A_8= \frac{A_7}{A_6}= \frac{1}{-0.5}= -2$$
• $$A_9=\frac{A_8}{A_7}= \frac{-2}{1}= -2$$
o If you see $$A_1= A_7= 1$$ , $$A_2= A_8=-2$$, and $$A_3= A_9=-2$$ then we can say that An repeats after every 6 number.
o Hence, $$A_{100} = A_ {6*16+4}= A_4= 1$$

Hence, the correct answer is option A.

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Re: If An=An-1/An-2, A1=1, and A2=-2, then A100=?  [#permalink]

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MathRevolution wrote:
[GMAT math practice question]

If $$A_n = \frac{A_{n-1}}{A_{n-2}}$$, $$A_1=1$$, and $$A_2 =-2$$, then $$A_{100}=$$?

$$A. 1$$

$$B. -1$$

$$C. 2$$

$$D. -2$$

$$E. \frac{-1}{2}$$

$$A_n = \frac{A_{n-1}}{A_{n-2}}............A_{n-1}=A_n*A_{n-2}$$
So the middle term is multiple of the side terms..
$$A_2=A_1*A_3.........-2=1*A_3........A_3=-2$$..
$$-2=-2*A_4......A_4=1$$..
$$1=-2*A_5.......A_5=\frac{-1}{2}$$..
$$\frac{-1}{2}=1*A_6......A_6=\frac{-1}{2}$$..
$$\frac{-1}{2}=\frac{-1}{2}*A_7......A_7=1$$...
So numbers are 1,-2,-2,1,-1/2,-1/2,1,-2.....
Thus sequence gets repeated after every 6 terms..
100=96+4=6*16+4
So 4th term of the sequence....1

A
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Re: If An=An-1/An-2, A1=1, and A2=-2, then A100=?  [#permalink]

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2^1=2
2^2=4
2^3=8
2^4=16
2^5=32
so, 2^30 the unit digit is 4
why same rule is not applicable for above question?
A0=-1/2
A1=1
.
.
.
.
A5=-1/2
A6=-1/2
A7=1
so, from A6 the value is repeating, Hence, (100/5)=20 times, means A100 must be -1/2
why it is A100=1 ???????????
kindly ans Plz.
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Re: If An=An-1/An-2, A1=1, and A2=-2, then A100=?  [#permalink]

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=>
A1 = 1, A2 =-2.
A3 = A2 / A1 = $$\frac{(-2)}{1} = -2$$.
A4 = A3 / A2 = $$\frac{(-2)}{(-2)} = 1.$$
A5 = A4 / A3 = $$\frac{1}{(-2)} = \frac{-1}{2}.$$
A6 = A5 / A4 = $$-(\frac{1}{2}) / 1 = \frac{-1}{2}.$$
A7 = A6 / A5 = $$-\frac{(1}{2)}$$ / $$-(\frac{1}{2})$$ = 1 = A1.
A8 = A7 / A6 = $$1 / (\frac{-1}{2}) = -2$$= 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 = … = $$-(\frac{1}{2}).$$
A6 = A12 = A18 = … = $$-(\frac{1}{2}).$$
Now, $$100 = 6 * 16 + 4.$$ Thus,
A100 = A4 = $$1$$.
_________________ Re: If An=An-1/An-2, A1=1, and A2=-2, then A100=?   [#permalink] 22 Jul 2018, 18:21
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