![]() ![]() ![]() But as Alan says, you could choose any value and make a polynomial to fit. So for the polynomial of lowest degree that generates the sequence as given, the next term would be #151#. It's somewhat easier than substituting #8# into the polynomial. In our case that does not happen, but see what happens when we add an extra row and reconstruct the next term of the sequence. If a sequence was really constructed using a polynomial then you would expect to encounter a row of #0#'s somewhat earlier, giving a simpler polynomial. We can use the first element of each row of differences (as enclosed by the green rectangle) to construct a polynomial for the sequence: This number d is called the common difference. Let's look at the differences and differences of differences. The next number, or term, in an arithmetic sequence is formed by adding a number to the previous term. So we use these to pick out each of the three rules cyclically.Īs Alan says, you can construct a polynomial of degree #n# for any sequence of #n# values. When #i=2# modulo #3#, then these coefficient expressions work out as #0#, #0# and #1#. When #i = 1# modulo #3#, then these coefficient expressions work out as #0#, #1# and #0#. This can be simplified, but it helps to have it in this formulation so you can understand how it works. Is there a single algebraic formula to describe this iterative process? This is not a very mathematically significant kind of sequence. ![]()
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