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

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The infinite sequence Sk is defined as Sk = 10 Sk – 1 + k, for all k > [#permalink]

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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
[Reveal] Spoiler: OA

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Last edited by abhimahna on 08 Oct 2017, 05:55, edited 1 time in total.
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The infinite sequence Sk is defined as Sk = 10 Sk – 1 + k, for all k > [#permalink]

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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.
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Re: The infinite sequence Sk is defined as Sk = 10 Sk – 1 + k, for all k > [#permalink]

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New post 07 Oct 2017, 05:01
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.

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Re: The infinite sequence Sk is defined as Sk = 10 Sk – 1 + k, for all k >   [#permalink] 07 Oct 2017, 05:01
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