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How do I find the next number if the given pattern is $$1,2,3,2,3,4,1,2,6,23,14,19,64,69,12,78,152,93,108,?$$ (Find the question mark)

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If there is no pattern you can put any number since it preserve the the structure – mesel Jun 11 '14 at 12:00
Fit a polynomial to it that satisfies $(1,1),(2,2),(3,3),...,(18,93),(19,108)$. Then plug in $20$ and BOOM! This works for any "pattern"! – AJ Stas Jun 11 '14 at 12:01
I think no need for downvote, it is a valid question. – mesel Jun 11 '14 at 12:03
Is this an exercise or this sequence arises from a mathematical / physical / cryptographic problem? In the first case, there is really no fixed answer but some random wild guesses. – achille hui Jun 11 '14 at 12:17
The question mark is at the end. – Joel Reyes Noche Jun 11 '14 at 13:16
up vote 12 down vote accepted

Let $p(n)$ be the $n$th term in the sequence. Clearly, this sequence follows the formula:

$$\begin{align} p(x) &= \frac{600631 x^{19}}{121645100408832000}-\frac{791723 x^{18}}{800296713216000}+\frac{196988587 x^{17}}{2134124568576000}\\ &-\frac{41785811 x^{16}}{7846046208000}+\frac{8219611 x^{15}}{38626689024}-\frac{49026370303 x^{14}}{7846046208000}+\frac{26296057821373 x^{13}}{188305108992000}\\ &-\frac{1098593289863 x^{12}}{452656512000}+\frac{320897391017407 x^{11}}{9656672256000}-\frac{39606777445183 x^{10}}{109734912000}\\ &+\frac{30088961291838131 x^9}{9656672256000}-\frac{(12871880314235441 x^8)}{603542016000}+\frac{(5410873671286319827 x^7)}{47076277248000}-\frac{(708875674839982733 x^6)}{1471133664000}+\frac{4033947669590964373 x^5)}{2615348736000}-\frac{(599274486262658993 x^4)}{163459296000} \\&\frac{+(49104110859304547 x^3)}{7916832000}-\frac{(2153634755170519 x^2)}{308756448}+\frac{(3188726258687 x)}{692835}-1320490 \end{align}$$

Thus, $p(20)=42$.

Ok, so that was a joke. However, this illustrates an important point--you can find some formula for $p(n)$ such that $p(20)$ is any value you wish. Without other context or information, coming up with the $20$th term in this sequence is not a well-defined question.

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That's brilliant! – Dmitry Kazakov Jun 11 '14 at 12:15
I thought there wouldn't be a formula to that cause the number looks rubbish.But this is a real question where can be solved – ministic2001 Jun 11 '14 at 12:16
This cannot be the way to get the answer because the formula has more than 42 characters. +1 anyways because we all bow down before the answer "42". – achille hui Jun 11 '14 at 12:22
Another way would be to use a Fourier interpolation and get another value. Mathematica with its default Hermite interpolation gives the result of 404. – Ruslan Jun 11 '14 at 12:24
@ministic2001 I am sure that it is a real question, but I am saying that the problem is not solvable as-is. We need context in order to provide a meaningful answer. – apnorton Jun 11 '14 at 12:25

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