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# If the sequence {An } satisfies An = An-1 - An-2, A1 = 0, and A2 = 1,

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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 6661
GMAT 1: 760 Q51 V42
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If the sequence {An } satisfies An = An-1 - An-2, A1 = 0, and A2 = 1,  [#permalink]

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12 Jul 2018, 00:12
00:00

Difficulty:

65% (hard)

Question Stats:

60% (02:45) correct 40% (03:10) wrong based on 53 sessions

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[GMAT math practice question]

If the sequence {An} satisfies An = An-1 - An-2, A1 = 0, and A2 = 1, where n is an integer greater than 2, then what is the sum of the first 100 terms of {An}?

A. 1
B. 2
C. 3
D. 4
E. 5

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"Only $99 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" MBA Section Director Affiliations: GMATClub Joined: 22 May 2017 Posts: 1539 Concentration: Nonprofit GPA: 4 WE: Engineering (Computer Software) Re: If the sequence {An } satisfies An = An-1 - An-2, A1 = 0, and A2 = 1, [#permalink] ### Show Tags 12 Jul 2018, 01:21 1 $$A_ 1 = 0$$ and $$A_ 2 = 1$$ $$A_ n = A_{n-1} - A_{n-2}$$ => $$A_ 3 = A_ 2 - A_ 1 = 1 - 0 = 1$$ => $$A_ 4 = A_ 3 - A_ 2 = 1 - 1 = 0$$ => $$A_ 5 = A_ 4 - A_ 3 = 0 - 1 = -1$$ => $$A_ 6 = A_ 5 - A_ 4 = -1 - 0 = -1$$ => $$A_ 7 = A_ 6 - A_ 5 = -1 - (-1) = -1 + 1 = 0$$ The sequence repeats 0, 1, 1, 0, -1, -1, 0, 1, 1 ..... The sequence repeats for every 6 terms and the sum of these six terms is 0 + 1 + 1 + 0 + (-1) + (-1) = 0 With in 100, there are 16 such groups till 96th term and the sum of all terms till 96th term is 0 => $$A_{97} = A_{96} - A_{95} = -1 - (-1) = 0$$ Similarly $$A_{98} = 1$$, $$A_{99} = 1$$, $$A_{100} = 0$$ Sum of first hundred terms is $$0 + A_{97} + A_{98} + A_{99} + A_{100} = 0 + 0 + 1 + 1 + 0 = 2$$ Hence option B _________________ Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6661 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: If the sequence {An } satisfies An = An-1 - An-2, A1 = 0, and A2 = 1, [#permalink] ### Show Tags 15 Jul 2018, 17:20 => We determine the period of the sequence by examining its terms: A1 = 0 and A2 = 1. A3 = A2 – A1 = 1 – 0 = 1 A4 = A3 – A2 = 1 – 1 = 0 A5 = A4 – A3 = 0 – 1 = -1 A6 = A5 – A4 = -1 – 0 = -1 A7 = A6 – A5 = -1 – (-1) = 0 A8 = A7 – A6 = 0 – (-1) = 1 A9 = A8 – A7 = 1 – 0 = 1 A10 = A9 – A8 = 1 – 1 = 0 The sequence has period 6. The sum of the first six terms is A1 + A2 + … A6 = 0 + 1 + 1 + 0 + (-1) + (-1) = 0. We apply this fact to determine the sum of the first 100 terms of the sequence: A1 + A2 + … A100 = ( A1 + A2 + … A6 ) + … + ( A91 + A92 + … A96 ) + A97 + A98 + A99 + A100 = 0 + … + 0 + A97 + A98 + A99 + A100 = A97 + A98 + A99 + A100 = 0 + 1 + 1 + 0 = 2. Therefore, the answer is B. _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$99 for 3 month Online Course"
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Re: If the sequence {An } satisfies An = An-1 - An-2, A1 = 0, and A2 = 1,  [#permalink]

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19 Jul 2018, 11:43
MathRevolution wrote:
[GMAT math practice question]

If the sequence {An} satisfies An = An-1 - An-2, A1 = 0, and A2 = 1, where n is an integer greater than 2, then what is the sum of the first 100 terms of {An}?

A. 1
B. 2
C. 3
D. 4
E. 5

Let’s list the first few terms to discern a pattern.

A1 = 0
A2 = 1
A3 = 1 - 0 = 1
A4 = 1 - 1 = 0
A5 = 0 - 1 = -1
A6 = -1 - 0 = -1
A7 = -1 - (-1) = 0
A8 = 0 - (-1) = 1
A9 = 1 - 0 = 1

At this point, we can see that the terms repeat themselves in a cycle of 6 numbers: 0, 1, 1, 0, -1, -1 (notice that A7 = A1, A8 = A2, A9 = A3, etc.). Also notice that the sum of the 6 numbers in one cycle is 0. So the sum of all the terms up to and including the 96th term is 0 (notice 96 = 6 x 16). So we really just need to add A97, A98, A99 and A 100. Since A97 = A1 = 0, A98 = A2 = 1, A99 = 1 and A100 = 0, the sum of these 4 terms (and hence the sum of the first 100 terms) is 0 + 1 + 1 + 0 = 2.

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Re: If the sequence {An } satisfies An = An-1 - An-2, A1 = 0, and A2 = 1, &nbs [#permalink] 19 Jul 2018, 11:43
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