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

Question

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

Bunuel · Accepted Answer

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.

Srisk · Answer

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.

anvesharma · Answer

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.

anvesharma · Answer

You're right, it's nowhere mentioned that n=k. Even if it weren't, the correct answer is still 18.

umangshah · Answer

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

Kinshook · Answer

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?

S_9 = 123456789; A_9 = 987654321; q = S_9 + A_9 = 1111111110; k+n = 9+9 = 18

IMO E

imaityaroy · Answer

Hi Experts
I have the same doubt as Srisk and umangshah. Can you please make me understand how 
Q1 = S1 + A1 
Q2 = S2+A2....and so on ?

Are we assuming n=k?

Why cant Q1 = S1 + A2 or any other combination?

I agree sum of digits will be equal to 9 when we add S9 and A9 
n+k =18 is the maximum value when we assume n=k
if n != k then does the above still hold true ? Is it too complicated to try out in under 2 mins ?

If the question had given n = k then the picture must have been clearer

Thanks

egmat · Answer

imaityaroy

Looking at your doubt, I can see you're overthinking the problem setup! Let me clarify this critical point that's causing confusion.

Key Clarification: k and n are INDEPENDENT variables

You're absolutely right to question this! The problem states that  q  is the sum of  S_k  and  A_n , where  k and n can be ANY positive integers  (with the constraint that  A_n  must be positive). The problem does  NOT  require  n = k .

Why the confusion? 
The problem asks for the  maximum  value of  k + n  when the digit sum of  q = S_k + A_n  equals  9 . You can choose ANY combination like:
 
  q = S_1 + A_8  
  q = S_2 + A_7  
  q = S_3 + A_6  
 ...and so on

The Strategic Insight 
Let's examine what happens with  k = n = 9 :
 
  S_9 = 123456789  (digits  1  through  9  in sequence) 
  A_9 = 987654321  (digits  9  through  1  in reverse) 
  q = 123456789 + 987654321 = 1111111110  
 Sum of digits =  1+1+1+1+1+1+1+1+1+0 = 9  ✓

This gives us  k + n = 9 + 9 = 18 .

Could we get a higher sum with different k and n? 
No! Here's why:
 
 If either  k > 9  or  n > 9 , we'd need  k + n > 18  to potentially beat our answer 
 But when  n > 9 , the sequence  A_n  starts including  0  and negative differences, making the pattern complex 
 When  k > 9 ,  S_k  requires multi-digit endings (like  ...10, 11, 12 ), disrupting the clean digit pattern

Testing other combinations quickly: 
You could try  k = 1, n = 17  to get  k + n = 18 , but:
 
  S_1 = 1  
  A_{17}  would have a complex pattern that likely doesn't give digit sum  9  when added to  1

Time Management Tip: 
The GMAT gives you this specific setup ( S_k  building  123...k  and  A_n  building  987...() ) precisely because when  k = n = 9 , you get the beautiful result  1111111110 . Once you spot this pattern, you can confidently choose  18  without checking all possibilities.

Answer:  The maximum value is  (E) 18 , achieved when  k = n = 9 , though the problem allows any valid  (k, n)  pair.

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

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