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Math Expert V
Joined: 02 Sep 2009
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Difficulty:   15% (low)

Question Stats: 76% (01:00) correct 24% (00:56) wrong based on 89 sessions

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A sequence is defined as follows:
$$A_1 = 1$$
$$A_2 = -1$$
$$A_{n+1} = A_n + 2A_{n-1}$$

What is the sum $$A_1 + A_2 + ... + A_{1001}$$?

A. -2
B. -1
C. 0
D. 1
E. 2

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Math Expert V
Joined: 02 Sep 2009
Posts: 58445

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Official Solution:

A sequence is defined as follows:
$$A_1 = 1$$
$$A_2 = -1$$
$$A_{n+1} = A_n + 2A_{n-1}$$

What is the sum $$A_1 + A_2 + ... + A_{1001}$$?

A. -2
B. -1
C. 0
D. 1
E. 2

Write out more terms of this sequence: $$A_3 = 1$$; $$A_4 = -1$$; $$A_5 = 1$$. Even terms equal -1 and odd terms equal 1. Therefore, the sum $$A_1 + A_2 + ... + A_{1001} = 1$$

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Intern  B
Joined: 05 Jan 2015
Posts: 2

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How did you get A3 = 1 , can you please show the setup ?
Math Expert V
Joined: 02 Sep 2009
Posts: 58445

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haris550 wrote:
How did you get A3 = 1 , can you please show the setup ?

$$A_1 = 1$$
$$A_2 = -1$$
$$A_{n+1} = A_n + 2A_{n-1}$$

According to the above: $$A_3 = A_2 + 2A_{1}=-1+2*1=1$$

Hope it's clear.
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Senior Manager  Joined: 31 Mar 2016
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Location: India
Concentration: Operations, Finance
GMAT 1: 670 Q48 V34 GPA: 3.8
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I think this is a high-quality question and I agree with explanation.
Intern  B
Joined: 09 Jul 2016
Posts: 17
GMAT 1: 730 Q50 V39 Show Tags

I think this is a high-quality question and I agree with explanation. This is surely a 600 Lv question
Intern  B
Joined: 20 May 2017
Posts: 1

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I am still confused after reading your explanation wouldn't the answer be -1?
Math Expert V
Joined: 02 Sep 2009
Posts: 58445

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POQBigwill wrote:
I am still confused after reading your explanation wouldn't the answer be -1?

Why do you think the answer would be -1? Hod did you get this?

The solutions says that even terms equal -1 and odd terms equal 1. We have the sum of odd number of terms (1001). The sum would be the same for any other odd number of terms: 1 + (-1) + 1 = 1.
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Manager  B
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Schools: ISB '20, NUS '21
GMAT 1: 420 Q26 V13 GMAT 2: 540 Q44 V21 Show Tags

Dear Experts chetan2u Bunuel

Even I got the answer -1 (I am sure I used the wrong formula and thinking)

When does the below formula hold true?
Sum of an infinite geometric progression with common ratio <1 is b/1-r (where b is the first term)

So (I may have wrongly assumed common ratio =-1-1 =-2 which is <1
hence according to the above formula
b/1-r = 1/(1-2) = -1
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Math Expert V
Joined: 02 Sep 2009
Posts: 58445

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deddex wrote:
Dear Experts chetan2u Bunuel

Even I got the answer -1 (I am sure I used the wrong formula and thinking)

When does the below formula hold true?
Sum of an infinite geometric progression with common ratio <1 is b/1-r (where b is the first term)

So (I may have wrongly assumed common ratio =-1-1 =-2 which is <1
hence according to the above formula
b/1-r = 1/(1-2) = -1

The sum of infinite geometric progression with common ratio $$|r|<1$$, is $$sum=\frac{b}{1-r}$$, where $$b$$ is the first term.

Take a look at the highlighted words.

1. We are not asked to find the sum of the infinite sum, we are asked to find the sum of the first 1,001 terms.
2. The sequence is: 1, -1, 1, -1, 1, .... While this IS in fact a geometric progression, the common ratio is -1 (you can find it by dividing any two consecutive terms). As you can see the absolute value of the ratio is not less than 1.

For more on sequences check below:

12. Sequences

For other subjects:
ALL YOU NEED FOR QUANT ! ! !
Ultimate GMAT Quantitative Megathread

Hope it helps.
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