Expert: Josh - 8/2/2008

Question
QUESTION: Please help with the next probable number:

4, 16, 32, 64

ANSWER: Dr Deepak

This is most unusual. It does not match a geometric sequence or a logarithmic profile. Are you sure there are no missing terms such as an "8" in between 4 and 16.

The following exponential second order equation will generate the numbers observed thus far.

Let T[0]=4, T[1]=16 and
T[n] = 2^(5-n) * T[n-2] for integer n>1

According to this recursive formula,
T[2] = 2^3 * 4 = 32
T[3] = 2^2 * 16 = 64
T[4] = 2^1 * 32 = 64 etc.

One can certain argue the case, but I don't think there is a definitive answer. This formula is not very appealing as it predicts a sequence that is not an increasing function.

Regards
Josh

---------- FOLLOW-UP ----------

QUESTION: Dear Josh,

Thanks a lot for your prompt response...However, the answer to the series 4, 16, 32, 64 is given as '300'...so the formula that you have suggested does not fit in...kindly see if you can now predict the formula/explanation..

Thanks

Regards
Deepak

Answer
With no a priori information, this is a hopeless task. One could come up with many guesses, ranging from linear models such as T[n]=4*T[n-1]+(T[n-2]+T[n-3]-T[n-4]) to ever more complicated ones.

As a patient cannot expect a specific diagnosis from his doctor complaining that he feels generally unwell, here we also need some guidance or context to the problem.

Knowing what process generated the numbers helps. Is the sequence deterministic? Is there a closed form solution? I wonder if it is worth the effort seeking the answer or answers. Finding a unique structure for the sequence is next to impossible given this little information.