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. 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. In an arithmetic progression the difference between one number and the next is always the same.
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