There are of course many more ways to construct sequences but the ones mentioned here are some of the most common. In addition to what has been mentioned already the tool can also recognize the sequence of prime numbers and the Fibonacci sequence. The reason the tool does not always find a polynomial has to do with technical limitations that makes the numeric precision not good enough for polynomials of higher degrees. This is something to think about when using the tool on this page. For this the polynomial degree would have to be two (preferable three or more) degrees lower than the number of known numbers in the sequence. If n numbers are known it is always possible to find a polynomial of degree n - 1 that match all the numbers, but this does not necessarily describe any true pattern of the sequence. Note that as long as you have a finite sequence of numbers it is always possible to find a polynomial that can describe it. For fourth degree polynomials we would have to look at yet another level of differences. To solve a third degree polynomial the difference between the differences between the differences need to be constant. Sometimes it can be necessary to use polynomials of higher degree than two but the method is essentially the same. To establish the polynomial we note that the formula will have the following form. This tells us that it is possible to describe the sequence as a second degree polynomial but it does not give us any information about how. If we look at the difference between the five initial numbers we find that they are 3 5 7 9 and, as you can see, the differences between these numbers are 2. 2 5 10 17 26… is an example of such a sequence. If it turns out that the difference between the differences is constant it means that the sequence can be described using a second degree polynomial. If neither quotient nor difference is constant it might be a good idea to look at the difference between the differences. This sequence can be described using the exponential formula a n = 2 n. 2 4 8 16… is an example of a geometric progression that starts with 2 and is doubled for each position in the sequence. In a geometric progression the quotient between one number and the next is always the same. This sequence can be described using the linear formula a n = 3 n − 2. ![]() and of course it does, so our formula that an+1 an + 10 works when n 1. ![]() 1 4 7 10 13… is an example of an arithmetic progression that starts with 1 and increases by 3 for each position in the sequence. An arithmetic sequence is a sequence a1,a2,a3. In an arithmetic progression the difference between one number and the next is always the same.
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