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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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Updated on: 08 Oct 2017, 05:55
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Difficulty:

95% (hard)

Question Stats:

47% (02:52) correct 53% (03:37) wrong based on 137 sessions

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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

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Originally posted by enigma123 on 13 Jan 2012, 00:04.
Last edited by abhimahna on 08 Oct 2017, 05:55, edited 1 time in total.
Edited the Question
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The infinite sequence Sk is defined as Sk = 10 Sk – 1 + k, for all k >  [#permalink]

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13 Jan 2012, 05:36
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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 --> $$k+n=9+9=18$$.

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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Joined: 03 Sep 2017
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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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07 Oct 2017, 05:01
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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.
Re: The infinite sequence Sk is defined as Sk = 10 Sk – 1 + k, for all k > &nbs [#permalink] 07 Oct 2017, 05:01
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