Manhattan Challenge Problem of the week ! - Jan 15

Sn = 2Sn-1 + 4 and Qn = 4Qn-1 + 8 for all n > 1. If S5 = Q4 and S7 = 316, what is the first value of n for which Qn is an integer?

(A) 1

(B) 2

(C) 3

(D) 4

(E) 5


Answer : C is the correct answer.

If S7 = 316, then 316 = 2S6 + 4, which means that S6=156.

We can then solve for S5:
156 = 2S5 + 4, so S5 = 76

Since S5 = Q4, we know that Q4 = 76 and we can now solve for previous Qn’s to find the first n value for which Qn is an integer.

To find Q3: 76 = 4Q3 + 8, so Q3 = 17
To find Q2: 17 = 4Q2 + 8, so Q2 = 9/2

It is clear that Q1 will also not be an integer so there is no need to continue.

Q3 (n = 3) is the first integer value.