Jul 26 08:00 AM PDT  09:00 AM PDT The Competition Continues  Game of Timers is a teambased competition based on solving GMAT questions to win epic prizes! Starting July 1st, compete to win prep materials while studying for GMAT! Registration is Open! Ends July 26th Jul 27 07:00 AM PDT  09:00 AM PDT Learn reading strategies that can help even nonvoracious reader to master GMAT RC Jul 28 07:00 PM EDT  08:00 PM EDT Strategies and techniques for approaching featured GMAT topics. Sunday, July 28th at 7 PM EDT
Author 
Message 
TAGS:

Hide Tags

Manager
Joined: 21 Jun 2010
Posts: 105
Schools: Tuck, Duke, Cambridge, Said

S is the infinite sequence S1=2, S2=22, S3=222
[#permalink]
Show Tags
18 Jul 2010, 15:47
Question Stats:
49% (02:44) correct 51% (02:42) wrong based on 479 sessions
HideShow timer Statistics
S is the infinite sequence S1 = 2, S2 = 22, S3 = 222,...Sk = Sk–1 + 2(10k–1). If p is the sum of the first 30 terms of S, what is the eleventh digit of p, counting right to left from the units digit? A) 1 B) 2 C) 4 D) 6 E) 9 Anyone knows what is the easiest and fastest method of solving this ?
Official Answer and Stats are available only to registered users. Register/ Login.




Math Expert
Joined: 02 Sep 2009
Posts: 56357

Re: Infinite seq of 2
[#permalink]
Show Tags
18 Jul 2010, 16:30
mn2010 wrote: S is the infinite sequence S1 = 2, S2 = 22, S3 = 222,...Sk = Sk–1 + 2(10k–1). If p is the sum of the first 30 terms of S, what is the eleventh digit of p, counting right to left from the units digit?
A) 1 B) 2 C) 4 D) 6 E) 9
Anyone knows what is the easiest and fastest method of solving this ? 2 22 222 2,222 22,222 ... 222,222,222,222,222,222,222,222,222,222 Total 30 numbers. For the first digit (units place) we should add 30 2's > 30*2=60, so 0 will be units digit and 6 will be carried over; For the second digit (tens place) we should add 29 2's > 29*2=58+6=64, so 4 will be written for this digit and 6 will be carried over; ... For the 10th digit we should add 21 2's > 21*2=42, so min value for the number carried over is 4. Max value is also 4, because even if the carry remained 6, as we had at the beginning, still > 42+6=48, so still 4 will be carried over; For the 11th digit we should add 20 2's > 20*2+4=44, so 11th digit will be 4. Answer: C. Just curios: is it GMAT question?
_________________




Director
Joined: 17 Feb 2010
Posts: 977

Re: Infinite seq of 2
[#permalink]
Show Tags
21 Jul 2010, 13:58
Hey Bunuel,
Is it possible to come up with a sequence formula for this question. Because we have the value (2) of first term and with the sequnece formula we can get the value of the last term (30th term).
Once we have the first and last term values, we can take the average of the first and last values and multiply that by 30 to get the sum of 30 terms. That way we can get the eleventh digit of p.
Please share your thoughts on the same.



Math Expert
Joined: 02 Sep 2009
Posts: 56357

Re: Infinite seq of 2
[#permalink]
Show Tags
21 Jul 2010, 14:21
seekmba wrote: Hey Bunuel,
Is it possible to come up with a sequence formula for this question. Because we have the value (2) of first term and with the sequnece formula we can get the value of the last term (30th term).
Once we have the first and last term values, we can take the average of the first and last values and multiply that by 30 to get the sum of 30 terms. That way we can get the eleventh digit of p.
Please share your thoughts on the same. Formula Sum = average of the first and last terms multiplied by # of terms can be used for evenly spaced set, but 2, 22, 222, ... is not such set, hence you can not use this formula here.
_________________



Director
Joined: 17 Feb 2010
Posts: 977

Re: Infinite seq of 2
[#permalink]
Show Tags
22 Jul 2010, 07:43
Thanks Bunuel. Makes sense. I forgot about the evenly spaced set and hence mixed it up.



Manager
Joined: 23 Oct 2011
Posts: 83

Re: Infinite seq of 2
[#permalink]
Show Tags
13 Jan 2012, 16:55
Bunuel wrote: mn2010 wrote: S is the infinite sequence S1 = 2, S2 = 22, S3 = 222,...Sk = Sk–1 + 2(10k–1). If p is the sum of the first 30 terms of S, what is the eleventh digit of p, counting right to left from the units digit?
A) 1 B) 2 C) 4 D) 6 E) 9
Anyone knows what is the easiest and fastest method of solving this ? 2 22 222 2,222 22,222 ... 222,222,222,222,222,222,222,222,222,222 Total 30 numbers. For the first digit (units place) we should add 30 2's > 30*2=60, so 0 will be units digit and 6 will be carried over; For the second digit (tens place) we should add 29 2's > 29*2=58+6=64, so 4 will be written for this digit and 6 will be carried over; ... For the 10th digit we should add 21 2's > 21*2=42, so min value for the number carried over is 4. Max value is also 4, because even if the carry remained 6, as we had at the beginning, still > 42+6=48, so still 4 will be carried over; For the 11th digit we should add 20 2's > 20*2+4=44, so 11th digit will be 4. Answer: C. Just curios: is it GMAT question? Another more risky solution would be to try and find a pattern.. 1st digit > 30*2=60 > 0 and 6 will be carried over 2nd digit > 29*2+6=64 > 4 and 6 will be carried over 3rd digit > 28*2+6=62 > 2 and 6 will be carried over 4th digit >27*2+6=60 > 0 and 6 will be carried over we could assume that this pattern will go on, therefore 11/3 gives as a remainder of 2 therefore the 11th digit will be the same wit the 2nd > C is my reasoning correct bunuel?



Math Expert
Joined: 02 Sep 2009
Posts: 56357

Re: Infinite seq of 2
[#permalink]
Show Tags
13 Jan 2012, 17:18
SonyGmat wrote: Bunuel wrote: mn2010 wrote: S is the infinite sequence S1 = 2, S2 = 22, S3 = 222,...Sk = Sk–1 + 2(10k–1). If p is the sum of the first 30 terms of S, what is the eleventh digit of p, counting right to left from the units digit?
A) 1 B) 2 C) 4 D) 6 E) 9
Anyone knows what is the easiest and fastest method of solving this ? 2 22 222 2,222 22,222 ... 222,222,222,222,222,222,222,222,222,222 Total 30 numbers. For the first digit (units place) we should add 30 2's > 30*2=60, so 0 will be units digit and 6 will be carried over; For the second digit (tens place) we should add 29 2's > 29*2=58+6=64, so 4 will be written for this digit and 6 will be carried over; ... For the 10th digit we should add 21 2's > 21*2=42, so min value for the number carried over is 4. Max value is also 4, because even if the carry remained 6, as we had at the beginning, still > 42+6=48, so still 4 will be carried over; For the 11th digit we should add 20 2's > 20*2+4=44, so 11th digit will be 4. Answer: C. Just curios: is it GMAT question? Another more risky solution would be to try and find a pattern.. 1st digit > 30*2=60 > 0 and 6 will be carried over 2nd digit > 29*2+6=64 > 4 and 6 will be carried over 3rd digit > 28*2+6=62 > 2 and 6 will be carried over 4th digit >27*2+6=60 > 0 and 6 will be carried over we could assume that this pattern will go on, therefore 11/3 gives as a remainder of 2 therefore the 11th digit will be the same wit the 2nd > C is my reasoning correct bunuel? Unfortunately not. If you continue you'll see that there is no pattern of 3: 5th digit > 26*2+6=58 > 8 and 5 will be carried over; 6th digit > 25*2+5=55 > 5 and 5 will be carried over; 7th digit > 24*2+5=53 > 3 and 5 will be carried over; 8th digit > 23*2+5=51 > 1 and 5 will be carried over; 9th digit > 22*2+5=49 > 9 and 4 will be carried over; 10th digit > 21*2+4=46 > 6 and 4 will be carried over; 11th digit > 20*2+4=44 > 4.
_________________



Senior Manager
Status: Finally Done. Admitted in Kellogg for 2015 intake
Joined: 25 Jun 2011
Posts: 462
Location: United Kingdom
Concentration: International Business, Strategy
GPA: 2.9
WE: Information Technology (Consulting)

Re: S is the infinite sequence S1=2, S2=22, S3=222
[#permalink]
Show Tags
13 Jan 2012, 23:12
Thanks Bunuel for a great and thorough explanation.
_________________
Best Regards, E.
MGMAT 1 > 530 MGMAT 2> 640 MGMAT 3 > 610 GMAT ==> 730



Manager
Status: MBA Aspirant
Joined: 12 Jun 2010
Posts: 133
Location: India
Concentration: Finance, International Business
WE: Information Technology (Investment Banking)

Re: S is the infinite sequence S1=2, S2=22, S3=222
[#permalink]
Show Tags
14 Jan 2012, 03:17
Bunuel thanks for the explanation



Senior Manager
Joined: 13 Aug 2012
Posts: 415
Concentration: Marketing, Finance
GPA: 3.23

Re: S is the infinite sequence S1=2, S2=22, S3=222
[#permalink]
Show Tags
21 Dec 2012, 08:06
1st digit: 2*30 = 60, carry over 6 2nd digit: 2*29 = 58 + 6 = 64, carry over 6 3rd: 2*28 = 56 + 6 = 62, carry over 6 4th: 2*27 = 54 + 6 = 60, carry over 6 5th: 2*26 = 52 + 6 = 58, carry over 5 6th: 2*25 = 50 + 5 = 55, carry over 5 7th: 2*24 = 48 + 5 = 53, carry over 5 8th: 2*23 = 46 + 5 = 51, carry over 5 9th: 2*22= 44 + 5 = 49, carry over 4 10th: 2*21 = 42 + 4 = 46, carry over 4 11th: 2*20 = 40 + 4 = 44 Answer: 4
_________________
Impossible is nothing to God.



Manager
Joined: 26 Jul 2011
Posts: 81
Location: India
WE: Marketing (Manufacturing)

Re: S is the infinite sequence S1=2, S2=22, S3=222
[#permalink]
Show Tags
21 Dec 2012, 08:35
bunuel
can you please explain this part " so min value for the number carried over is 4. Max value is also 4, because even if the carry remained 6, as we had at the beginning, still > 42+6=48, so still 4 will be carried over"



Intern
Joined: 17 Dec 2012
Posts: 3

Re: S is the infinite sequence S1=2, S2=22, S3=222
[#permalink]
Show Tags
21 Dec 2012, 17:28
Can someone please clarify what use is the "Sk = Sk–1 + 2(10k–1)" in the stem? is it used to throw you off? I'm not clear as to the reason it was provided nor do I see how it's used to provide the answer, 4.
I understand why the answer is 4, just not why or what that expression is provided for.



Math Expert
Joined: 02 Sep 2009
Posts: 56357

Re: S is the infinite sequence S1=2, S2=22, S3=222
[#permalink]
Show Tags
23 Dec 2012, 07:15
ratinarace wrote: bunuel
can you please explain this part " so min value for the number carried over is 4. Max value is also 4, because even if the carry remained 6, as we had at the beginning, still > 42+6=48, so still 4 will be carried over" For the 10th digit we should add 21 2's > 21*2=42. Even if carry from the previous operation is 6 (max possible), the sum would be 42+6=48 and the carried over for the next operation will be 4.
_________________



Math Expert
Joined: 02 Sep 2009
Posts: 56357

Re: S is the infinite sequence S1=2, S2=22, S3=222
[#permalink]
Show Tags
23 Dec 2012, 07:18
xchakax wrote: Can someone please clarify what use is the "Sk = Sk–1 + 2(10k–1)" in the stem? is it used to throw you off? I'm not clear as to the reason it was provided nor do I see how it's used to provide the answer, 4.
I understand why the answer is 4, just not why or what that expression is provided for. Formula \(S_k = S_{k1} + 2*10^{k1}\) gives numbers of the sequence: 2, 22, 222, 2222, ...
_________________



Intern
Joined: 21 Jun 2012
Posts: 3

Re: S is the infinite sequence S1=2, S2=22, S3=222
[#permalink]
Show Tags
24 Dec 2012, 22:25
mn2010 wrote: S is the infinite sequence S1 = 2, S2 = 22, S3 = 222,...Sk = Sk–1 + 2(10k–1). If p is the sum of the first 30 terms of S, what is the eleventh digit of p, counting right to left from the units digit?
A) 1 B) 2 C) 4 D) 6 E) 9
Anyone knows what is the easiest and fastest method of solving this ? The answer is C i.e. 4. The 11th place from left has twenty 2's. Sum of which is 40, note there is a carry of 4 from the 10th digit from left and thus the total sum is 44. Thus the 11th digit from left is 4. I have used the long conventional way to figure this out. It took a good 3 mins. Can anyone share a way where you can find trends in the data. Thanks



Senior Manager
Joined: 28 Apr 2012
Posts: 275
Location: India
Concentration: Finance, Technology
GMAT 1: 650 Q48 V31 GMAT 2: 770 Q50 V47
WE: Information Technology (Computer Software)

Re: S is the infinite sequence S1 = 2, S2 = 22, S3 = 222,...Sk =
[#permalink]
Show Tags
08 Jun 2013, 01:56
here is a good manual solution and it works... sistheinfinitesequences12s222s3222s33128.html#p227101ps_dahiya wrote: C
Sum of unit digits of first 30 terms = 60 Sum of tens digits of first 30 terms = 58 Sum of thousands digits of first 30 terms = 56 and so on..
p1 = 0 p2 = (6+58) = 4 p3 = (6+56) = 2 p4 = (6+54) = 0 p5 = (6+52) = 8 p6 = (5+50) = 5 p7 = (5+48) = 3 p8 = (5+46) = 1 p9 = (5+44) = 9 p10= (4+42) = 6 p11= (4+40) = 4: ANSWER Carry over is added to next sum.
_________________
"Appreciation is a wonderful thing. It makes what is excellent in others belong to us as well." ― Voltaire
Press Kudos, if I have helped. Thanks!



Intern
Joined: 02 May 2013
Posts: 20
Concentration: International Business, Technology
WE: Engineering (Aerospace and Defense)

Re: S is the infinite sequence S1 = 2, S2 = 22, S3 = 222,...Sk =
[#permalink]
Show Tags
08 Jun 2013, 02:07
S = 2,22,222,2222 ........ 30 terms Sum = 2+22+222+2222+.........+30 terms Let unit digit of sum = 2* 30 =60 (zero is unit digit) Let Tens digit = 2* 29 +6(carry) = 64 (4 is tens digit)
To find 11th digit from right end we do 2*20 + x(carry from 10th digit) = 40 +x
To find x lets find 10th digit= 2*21 +y(carry from 9th digit) For Eg: unit digit got carry = 6, then y should be less than/equal to 6 therefore 10th digit be 42+6 or 42 +5 or (42 + some positive number less than/equal to 6). So for 11th digit carry be 4 Then 11th digit be (40 + 4 =44) = 4
OA = C(4)



Intern
Joined: 11 Sep 2013
Posts: 20

Re: S is the infinite sequence S1=2, S2=22, S3=222
[#permalink]
Show Tags
07 Sep 2018, 11:35
2 22 222 2222 22222 222222 2222222 22222222 222222222 2222222222 22222222222 41975308642
Hence, the eleventh digit of p, counting right to left from the units digit is 4 (C)




Re: S is the infinite sequence S1=2, S2=22, S3=222
[#permalink]
07 Sep 2018, 11:35






