The sequence of numbers {Q(m)} is defined recursively by the following relationships:

Q(1) = 1, and:
Q(m) = 1 + Q(m/2), whenever m is ≥ 2 and even, and:
Q(m) = 1/Q(m-1), whenever m is ≥ 3 and odd.

Determine the value of d, given that Q(d) = 19/87

Strange as it may seem, it will prove useful to write 19/87 as

1/(1 + 3 + 1/(1 + 1/(1 + 1/(1 + 1+ 1/(1 + 1/(1 + 1))))))

Next note that

Q(2^N [2M + 1]) = N + 1/(1 + Q(M))

But

1905 = 2 * 8 * (2 * (2 * (4 * (2 * (2 + 1) + 1) + 1) + 1) + 1 )

So Q(1905) = 19/87

 

Posted by FrankM on 2008-02-16 12:11:57