Infinite Sequence

What is the source of the question, it did take about 3-4 mins for me, don't know about Math Gurus.

q is equal to An + Sn. And we are given S1 = 1 and A1 = 9, so S1 + A1 = 10.. sum of digits = 1 (1+0)

Further solving we find S2 = 10S1 + 2 and A2 = 11A1 - 1, so S2 + A2 = 10*1 + 2 + 11*9 - 1 = 110.. sum of digits = 2 (1+1+0)

Similarly S3 = 10S2 + 3 and A3 = 10A2 + A1 - 2, so S3 + A3 = 10*12 + 3 + 10*98 + 9 - 2 = 1110.. sum of digits = 3 (1+1+1+0)

So a pattern does emerge....hence for sum of digits to be 9, we have to have k+n = 9 + 9 = 18.

Hope this helps!!!

Another method is:-

S sequence is like 1, 12, 123 ,1234.....
A sequence is like 9, 98, 987 ,9876.....

So sum of (S + A) digits is 1, 2, 3, 4.....Hence 9th terms of both S and A will yield the sum as 9.

So, 9+9 = 18 is the answer.

Thanks for your response. This problem is from a set of questions that I found on internet. It claims to contain old actual GMAT questions.