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# a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divided by 3 f

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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 6967
GMAT 1: 760 Q51 V42
GPA: 3.82
a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divided by 3 f  [#permalink]

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08 Feb 2019, 01:42
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Difficulty:

65% (hard)

Question Stats:

42% (03:05) correct 58% (03:09) wrong based on 12 sessions

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[GMAT math practice question]

a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divided by $$3$$ for $$n ≥ 2$$. What is the value of a101 + a102 + a103 + a104 + a105 + a106 + a107 + a108?

A. 0
B. 3
C. 5
D. 7
E. 9

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MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
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"Only $149 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Manager Joined: 09 Jun 2014 Posts: 238 Location: India Concentration: General Management, Operations Schools: Tuck '19 a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divided by 3 f [#permalink] ### Show Tags Updated on: 08 Feb 2019, 03:51 MathRevolution wrote: [GMAT math practice question] a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divided by $$3$$ for $$n ≥ 2$$. What is the value of a101 + a102 + a103 + a104 + a105 + a106 + a107 + a108? A. 0 B. 3 C. 5 D. 7 E. 9 This is cyclicity question : The series is as below A1- 1 A2- 1 A3- 2 A4- 0 A5- 2 A-6 2 A7- 1 A8- 0 So together the sum of first segments gives 9 ..Now picking up any random segments will lead to same set of digits which will sum up to 9 So for even A161+ A162+ ...A168 =sum of digits will be 9 . or A3 to A11 Hope it helps!! Originally posted by prabsahi on 08 Feb 2019, 02:52. Last edited by prabsahi on 08 Feb 2019, 03:51, edited 2 times in total. VP Joined: 09 Mar 2018 Posts: 1001 Location: India Re: a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divided by 3 f [#permalink] ### Show Tags 08 Feb 2019, 03:18 1 MathRevolution wrote: [GMAT math practice question] a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divided by $$3$$ for $$n ≥ 2$$. What is the value of a101 + a102 + a103 + a104 + a105 + a106 + a107 + a108? A. 0 B. 3 C. 5 D. 7 E. 9 The trick was to find a pattern. Keyword = Remainder when divided by 3 an is the remainder when an-1 + an-2 is divided by $$3$$ a_0 = 0 a_1 = 1 a_2 = 1 (1+0)/3 a_3 = 2 (1+1)/3 a_4 = 0 (2+1)/3 a_5 = 2 (0+2)/3 a_6 = 2 (0+2)/3 a_7 = 1 (2+2)/3 a_8 = 0(2+1)/3 If you go forward you will notice the same series. So now the series starts from 101 ...... 108 1+1+2+0+2+2+1+0 9 E _________________ If you notice any discrepancy in my reasoning, please let me know. Lets improve together. Quote which i can relate to. Many of life's failures happen with people who do not realize how close they were to success when they gave up. Manager Joined: 09 Jun 2014 Posts: 238 Location: India Concentration: General Management, Operations Schools: Tuck '19 Re: a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divided by 3 f [#permalink] ### Show Tags 08 Feb 2019, 03:32 KanishkM wrote: MathRevolution wrote: [GMAT math practice question] a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divided by $$3$$ for $$n ≥ 2$$. What is the value of a101 + a102 + a103 + a104 + a105 + a106 + a107 + a108? A. 0 B. 3 C. 5 D. 7 E. 9 The trick was to find a pattern. Keyword = Remainder when divided by 3 an is the remainder when an-1 + an-2 is divided by $$3$$ a_0 = 0 a_1 = 1 a_2 = 1 (1+0)/3 a_3 = 2 (1+1)/3 a_4 = 0 (2+1)/3 a_5 = 2 (0+2)/3 a_6 = 2 (0+2)/3 a_7 = 1 (2+2)/3 a_8 = 0(2+1)/3 If you go forward you will notice the same series. So now the series starts from 101 ...... 108 1+1+2+0+2+2+1+0 9 E I see .I made a mistake Thanks VP Joined: 09 Mar 2018 Posts: 1001 Location: India Re: a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divided by 3 f [#permalink] ### Show Tags 08 Feb 2019, 03:38 prabsahi wrote: I see .I made a mistake Thanks prabsahi, your approach was right. But i still feel that is not the easiest way to solve it. _________________ If you notice any discrepancy in my reasoning, please let me know. Lets improve together. Quote which i can relate to. Many of life's failures happen with people who do not realize how close they were to success when they gave up. Manager Joined: 09 Jun 2014 Posts: 238 Location: India Concentration: General Management, Operations Schools: Tuck '19 Re: a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divided by 3 f [#permalink] ### Show Tags 08 Feb 2019, 03:44 KanishkM wrote: prabsahi wrote: I see .I made a mistake Thanks prabsahi, your approach was right. But i still feel that is not the easiest way to solve it. yes..You are right KanishkM..Will update it Thanks SVP Joined: 18 Aug 2017 Posts: 1887 Location: India Concentration: Sustainability, Marketing GPA: 4 WE: Marketing (Energy and Utilities) Re: a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divided by 3 f [#permalink] ### Show Tags 08 Feb 2019, 04:11 MathRevolution wrote: [GMAT math practice question] a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divided by $$3$$ for $$n ≥ 2$$. What is the value of a101 + a102 + a103 + a104 + a105 + a106 + a107 + a108? A. 0 B. 3 C. 5 D. 7 E. 9 the pattern here followed is of divisibility of a no with 3 ; which is always for consective no is series of 0,1,2,0,1,2 0,1,2 is our pattern so the 99th term would be 2 and 101st would be 1 so sum = 1+2+0+1+2+0+1+2; 9 IMO E _________________ If you liked my solution then please give Kudos. Kudos encourage active discussions. Manager Joined: 09 Jun 2014 Posts: 238 Location: India Concentration: General Management, Operations Schools: Tuck '19 a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divided by 3 f [#permalink] ### Show Tags 08 Feb 2019, 04:14 Archit3110 wrote: MathRevolution wrote: [GMAT math practice question] a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divided by $$3$$ for $$n ≥ 2$$. What is the value of a101 + a102 + a103 + a104 + a105 + a106 + a107 + a108? A. 0 B. 3 C. 5 D. 7 E. 9 the pattern here followed is of divisibility of a no with 3 ; which is always for consective no is series of 0,1,2,0,1,2 0,1,2 is our pattern so the 99th term would be 2 and 101st would be 1 so sum = 1+2+0+1+2+0+1+2; 9 IMO E Hi Archit, I guess you made a similar mistake like I did. Please check the pattern mentioned in my previous post or KanishkM's.I have corrected it. Its 0 1 1 2 0 2 2 1 0 Hope it helps !! Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6967 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divided by 3 f [#permalink] ### Show Tags 10 Feb 2019, 17:14 => a0 = 0 a1 = 1 a2 = a1 + a0 = 1 + 0 = 1 a3 = a2 + a1 = 1 + 1 = 2 a4 = 0 since a3 + a2 = 2 + 1 = 3 = 3(1)+0, and the remainder when a3 + a2 is divided by 3 is zero. a5 = a4 + a3 = 0 + 2 = 2 a6 = a5 + a4 = 2 + 0 = 2 a7 = 1 since a6 + a5 = 2 + 2 = 4= 3(1)+1, and the remainder when a6 + a5 is divided by 3 is 1. a8 = 0 since a7 + a6 = 1 + 2 = 3= 3(1)+0, the remainder is 0, when a7 + a6 is divided by 3. a9 = a8 + a7 = 0 + 1 = 1 Thus, the sequence is periodic, with period 8. a101 + a102 + a103 + a104 + a105 + a106 + a107 + a108 = a5 + a6 + a7 + a0 + a1 + a2 + a3 + a4 $$= 2 + 2 + 1 + 0 + 1 + 1 + 2 + 0 = 9$$ Therefore, the answer is E. Answer: E _________________ 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$149 for 3 month Online Course"
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Joined: 13 Mar 2018
Posts: 10
Re: a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divided by 3 f  [#permalink]

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13 Feb 2019, 06:51
KanishkM wrote:
MathRevolution wrote:
[GMAT math practice question]

a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divided by $$3$$ for $$n ≥ 2$$. What is the value of a101 + a102 + a103 + a104 + a105 + a106 + a107 + a108?

A. 0
B. 3
C. 5
D. 7
E. 9

The trick was to find a pattern.
Keyword = Remainder when divided by 3
an is the remainder when an-1 + an-2 is divided by $$3$$

a_0 = 0
a_1 = 1
a_2 = 1 (1+0)/3
a_3 = 2 (1+1)/3
a_4 = 0 (2+1)/3
a_5 = 2 (0+2)/3
a_6 = 2 (0+2)/3
a_7 = 1 (2+2)/3
a_8 = 0(2+1)/3

If you go forward you will notice the same series.

So now the series starts from 101 ...... 108
1+1+2+0+2+2+1+0
9

E

I dont understand what quotient to take here. Please explain.
Re: a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divided by 3 f   [#permalink] 13 Feb 2019, 06:51
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# a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divided by 3 f

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