A person is playing a video game where the number of aliens that appear on a level triples each time from the previous level. Le
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A plausible guess might be that the sequence is formed by a degree-4* polynomial,
From the given known values of the sequence, we have
Solving the system yields coefficients
so that the n-th term in the sequence might be
Then the next few terms in the sequence could very well be
It would be much easier to confirm this had the given sequence provided just one more term...
* Why degree-4? This rests on the assumption that the higher-order forward differences of eventually form a constant sequence. But we only have enough information to find one term in the sequence of 4th-order differences. Denote the k-th-order forward differences of
by
. Then
• 1st-order differences:
• 2nd-order differences:
• 3rd-order differences:
• 4th-order differences:
From here I made the assumption that is the constant sequence {15, 15, 15, …}. This implies
forms an arithmetic/linear sequence, which implies
forms a quadratic sequence, and so on up
forming a quartic sequence. Then we can use the method of undetermined coefficients to find it.