Last visit was: 08 Jun 2024, 04:09 It is currently 08 Jun 2024, 04:09
Decision Tracker
My Rewards
Unanswered
Active Topics
New posts
New comers' posts
Close
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Go to My Error Log Learn more
Close
Request Expert Reply
Confirm Cancel

The infinite sequence Sk is defined as Sk = 10 Sk 1 + k, for all k >

SORT BY:
Date
Kudos
Tags:
Find Similar Topics 
Show Tags
Hide Tags
User avatar
enigma123
Senior Manager
Senior Manager
Joined: 25 Jun 2011
Status:Finally Done. Admitted in Kellogg for 2015 intake
Posts: 395
Own Kudos [?]: 16853 [38]
Given Kudos: 217
Location: United Kingdom
Concentration: International Business, Strategy
GMAT 1: 730 Q49 V45
GPA: 2.9
WE:Information Technology (Consulting)
Send PM
The infinite sequence Sk is defined as Sk = 10 Sk 1 + k, for all k > [#permalink] The infinite sequence Sk is defined as Sk = 10 Sk 1 + k, for all k >   Updated on: 01 Nov 2023, 23:56
3
Kudos
35
Bookmarks
00:00
A
B
C
D
E

Difficulty:

95% (hard)

Question Stats:

40% (03:12) correct 60% (03:29) wrong based on 219 sessions
Hide Show DIFFICULTY AND TIMER STATISTICS
The infinite sequence \(S_k\) is defined as \(S_k=10*S_{k-1}+k\), for all k > 1. The infinite sequence \(A_n\) is defined as \(A_n=10*A_{n-1}+A_1-(n-1)\), for all n > 1. q is the sum of \(S_k\) and \(A_n\). If \(S_1 = 1\) and \(A_1 = 9\), and if \(A_n\) is positive, what is the maximum value of k + n when the sum of the digits of q is equal to 9?

(A) 6
(B) 9
(C) 12
(D) 16
(E) 18
ShowHide Answer
Official Answer
E

Originally posted by enigma123 on 13 Jan 2012, 00:04.
Last edited by Bunuel on 01 Nov 2023, 23:56, edited 2 times in total.
Most Helpful Reply
User avatar
L
Bunuel
Math Expert
Joined: 02 Sep 2009
Posts: 93603
Own Kudos [?]: 629991 [17]
Given Kudos: 82186
Send PM
CAT Tests Forum Quiz Mode
Re: The infinite sequence Sk is defined as Sk = 10 Sk 1 + k, for all k > [#permalink] The infinite sequence Sk is defined as Sk = 10 Sk 1 + k, for all k >   13 Jan 2012, 05:36
9
Kudos
8
Bookmarks
Expert Reply
enigma123 wrote:
The infinite sequence Sk is defined as Sk = 10 Sk – 1 + k, for all k > 1. The infinite sequence An is defined as An = 10 An – 1 + (A1 – (n - 1)), for all n > 1. q is the sum of Sk and An. If S1 = 1 and A1 = 9, and if An is positive, what is the maximum value of k + n when the sum of the digits of q is equal to 9?
(A) 6
(B) 9
(C) 12
(D) 16
(E) 18

Guys - any idea how to solve this? I am really struggling without the OA. Therefore your help will be very much appreciated.


For such kind of sequence problems when the formula of \(n_{th}\) term is given it's almost always a good idea to write down first few terms.

Given: \(S_k=10*S_{k-1}+k\) and \(A_n=10*A_{n-1}+A_1-(n-1)\);
\(S_1=1\) and \(A_1=9\);
\(S_2=10*1+2=12\) and \(A_2=10*9+9-(2-1)=98\);
\(S_3=10*12+3=123\) and \(A_3=10*98+9-(3-1)=987\);

\(Q_1=S_1+A_1=1+9=10\) - the sum of the digit of Q is 1;
\(Q_2=S_2+A_2=12+98=110\) - the sum of the digit of Q is 2;
\(Q_3=S_1+A_3=123+987=1,110\) - the sum of the digit of Q is 3;
...

We can see the pattern in \(Q_n\): the sum of its digit equals to \(n\) itself. So the first Q for which the sum of its digit is multiple of 9 is for \(Q_9=S_9+A_9\) --> sum of the digits of \(Q_9\) is 9 --> \(k+n=9+9=18\).

Answer: E.

P.S. I wonder whether the question supposed to ask the minimum value of k + n when the sum of the digits of q is equal to 9, though anyway as 18 is the the largest value from among the answer choices it's still the right one.
General Discussion
avatar
Srisk
Intern
Intern
Joined: 03 Sep 2017
Posts: 2
Own Kudos [?]: 1 [1]
Given Kudos: 10
Send PM
Re: The infinite sequence Sk is defined as Sk = 10 Sk 1 + k, for all k > [#permalink] The infinite sequence Sk is defined as Sk = 10 Sk 1 + k, for all k >   07 Oct 2017, 05:01
1
Kudos
Hello Bunuel,

Thanks for the detailed explanation.

Just one doubt wrt this qn. In the qn it states "q is the sum of Sk and An". Nowhere does it say n=k. Thus, q1 could also be a1+s2, correct?
Your solution is based on the assumption that n=k. Do help me understand the qn better in case I am missing something.

Thanks.
avatar
B
anvesharma
Intern
Intern
Joined: 16 Jul 2017
Posts: 3
Own Kudos [?]: 1 [0]
Given Kudos: 7
Send PM
Re: The infinite sequence Sk is defined as Sk = 10 Sk 1 + k, for all k > [#permalink] The infinite sequence Sk is defined as Sk = 10 Sk 1 + k, for all k >   19 Nov 2019, 14:38
Bunuel wrote:
enigma123 wrote:
The infinite sequence Sk is defined as Sk = 10 Sk – 1 + k, for all k > 1. The infinite sequence An is defined as An = 10 An – 1 + (A1 – (n - 1)), for all n > 1. q is the sum of Sk and An. If S1 = 1 and A1 = 9, and if An is positive, what is the maximum value of k + n when the sum of the digits of q is equal to 9?
(A) 6
(B) 9
(C) 12
(D) 16
(E) 18

Guys - any idea how to solve this? I am really struggling without the OA. Therefore your help will be very much appreciated.


For such kind of sequence problems when the formula of \(n_{th}\) term is given it's almost always a good idea to write down first few terms.

Given: \(S_k=10*S_{k-1}+k\) and \(A_n=10*A_{n-1}+A_1-(n-1)\);
\(S_1=1\) and \(A_1=9\);
\(S_2=10*1+2=12\) and \(A_2=10*9+9-(2-1)=98\);
\(S_3=10*12+3=123\) and \(A_3=10*98+9-(3-1)=987\);

\(Q_1=S_1+A_1=1+9=10\) - the sum of the digit of Q is 1;
\(Q_2=S_2+A_2=12+98=110\) - the sum of the digit of Q is 2;
\(Q_3=S_1+A_3=123+987=1,110\) - the sum of the digit of Q is 3;
...

We can see the pattern in \(Q_n\): the sum of its digit equals to \(n\) itself. So the first Q for which the sum of its digit is multiple of 9 is for \(Q_9=S_9+A_9\) --> sum of the digits of \(Q_9\) is 9 --> \(k+n=9+9=18\).

Answer: E.

P.S. I wonder whether the question supposed to ask the minimum value of k + n when the sum of the digits of q is equal to 9, though anyway as 18 is the the largest value from among the answer choices it's still the right one.



The minimum value of k+n should be 9. A1+S2=21, A2+S3=221, A3+S4=2221 and A4+S5=22221. So min value of k+n =9. This would also have been the correct answer choice had the question additionally mentioned that n is even.
avatar
B
anvesharma
Intern
Intern
Joined: 16 Jul 2017
Posts: 3
Own Kudos [?]: 1 [0]
Given Kudos: 7
Send PM
Re: The infinite sequence Sk is defined as Sk = 10 Sk 1 + k, for all k > [#permalink] The infinite sequence Sk is defined as Sk = 10 Sk 1 + k, for all k >   19 Nov 2019, 14:40
Srisk wrote:
Hello Bunuel,

Thanks for the detailed explanation.

Just one doubt wrt this qn. In the qn it states "q is the sum of Sk and An". Nowhere does it say n=k. Thus, q1 could also be a1+s2, correct?
Your solution is based on the assumption that n=k. Do help me understand the qn better in case I am missing something.

Thanks.


You're right, it's nowhere mentioned that n=k. Even if it weren't, the correct answer is still 18.
User avatar
S
umangshah
Manager
Manager
Joined: 24 May 2023
Status:Preparing for Exams
Posts: 71
Own Kudos [?]: 59 [0]
Given Kudos: 107
Location: India
GMAT 1: 640 Q44 V34
GPA: 4
WE:Consulting (Consulting)
Send PM
Re: The infinite sequence Sk is defined as Sk = 10 Sk 1 + k, for all k > [#permalink] The infinite sequence Sk is defined as Sk = 10 Sk 1 + k, for all k >   01 Nov 2023, 22:44
Srisk wrote:
Hello Bunuel,

Thanks for the detailed explanation.

Just one doubt wrt this qn. In the qn it states "q is the sum of Sk and An". Nowhere does it say n=k. Thus, q1 could also be a1+s2, correct?
Your solution is based on the assumption that n=k. Do help me understand the qn better in case I am missing something.

Thanks.


Hello Bunuel,

Can you clear the above. I too have the same doubt. It is true that we can solve the problem by assuming n=k, but i suppose there's also got to be a way without doing that. Please share you opinion on this.

thanks
GMAT Club Bot
gmatclubot
Re: The infinite sequence Sk is defined as Sk = 10 Sk 1 + k, for all k > [#permalink]
01 Nov 2023, 22:44
Moderator:
Bunuel
Math Expert
93602 posts