Some sequences are Geometric Progressions: $a_0, a_0+k, a_0+k^2, a_0+k^3. Some sequences are Arithmetic Progressions: $a_0, a_0+k, a_0+2k.$ There are some rules of thumb for sorting out sequences like this. (We can probably read C off of our table of first differences.) Now we need to know what has a first difference of n? Well, what do you get if you add the first n integers? n(n+1)/2 of course, so that's always the anti-difference of n, and the terms will be n(n+1)/2 + Cn + D. So if we're working with a sequence (this works just as well for a function), the formula for the first difference is n + C. Then the first difference (divided by the interval) is n and, just like integrating, we have to add a constant. (At this stage, that's probably a constant.) Then you go backward. People went hog-wild with this stuff before computers, and then it just disappeared in an instant.)Īnyway, you keep taking differences until you get a pattern you recognize. There's also a central difference, with a minuscule delta, and even off-center differences. I haven't actually done this stuff since maybe 1985. There is one difference from taking derivatives: we could also look at a(i-1) - a(i-2) = del(a(i-1)) = the backwards difference. If you take the forward difference of the forward difference, that's the second difference. It's a finite difference, the finite analog of the numerator when taking the derivative. You would write down your sequence in a column (maybe with the indices in the column to the left, so you don't get lost), and then in the spaces between the a(i) and to their right, you would write a(i) - a(i-1) = delta(a(i-1)). doi: 10.1511/2006.59.200.If you look at "really old" numerical analysis books (before about, maybe, 1970), they almost always had a section on (at least) forward differences. Polynomials calculating sums of powers of arithmetic progressions.Problems involving arithmetic progressions.Heronian triangles with sides in arithmetic progression.Generalized arithmetic progression, a set of integers constructed as an arithmetic progression is, but allowing several possible differences.Inequality of arithmetic and geometric means.However, the intersection of infinitely many infinite arithmetic progressions might be a single number rather than itself being an infinite progression. If each pair of progressions in a family of doubly infinite arithmetic progressions have a non-empty intersection, then there exists a number common to all of them that is, infinite arithmetic progressions form a Helly family. The intersection of any two doubly infinite arithmetic progressions is either empty or another arithmetic progression, which can be found using the Chinese remainder theorem. The formula is very similar to the standard deviation of a discrete uniform distribution. If the initial term of an arithmetic progression is a 1 is the common difference between terms. is an arithmetic progression with a common difference of 2. The constant difference is called common difference of that arithmetic progression. In order to check whether a term is or is not in the sequence, we set the nth term formula equal to the number that may or may not lie in the sequence. An arithmetic progression or arithmetic sequence ( AP) is a sequence of numbers such that the difference from any succeeding term to its preceding term remains constant throughout the sequence.
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