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Progressions
 >>fast Functions step ratio sum of the terms in a progression

WIRIS detects if a sequence of numbers, given its first few terms, follows a constant, arithmetic, geometric or polynomial progression. This allows the general term of a sequence to be determined or its terms to be summed using the familiar formulae. The command progression allows us to determine which type of progression a sequence of numbers follows.

WIRIS classifies progressions in accordance with the order in which we have just listed them. So, if a progression is constant, it classifies it as constant, even though it is also arithmetic and geometric. Equally, an arithmetic progression, which corresponds to a first degree polynomial, is categorised as arithmetic.

For every finite sequence of n numbers, there is a single polynomial of degree no higher than n-1, such that the first n terms in the corresponding polynomial sequence coincide with those of the sequence. WIRIS always forms the polynomial sequence corresponding to the polynomial of the lowest degree that meets this condition.

Once the progression is defined, we can save it in a variable. If p is a variable, then the expression p(i) returns its ith term for any number i, and if n is a variable, the expression p(n) returns the formula for the general term of the progression.

The functions associated with progressions are:

step:  command step

Given an arithmetic progression, it obtains a step (that is, the difference between two terms). If working with a constant progression, the function returns the value 0.

ratio:  command ratio

Given a geometric progression, its ratio is calculated. If working with a constant progression, the function returns the value 1.

sum of the terms in a progression:  command progression_sigma

Given a progression, the sum of its terms is obtained. Note that the result does not always have the appearance usually associated with a sum, due to the generality of the methods used. Nonetheless, the value of the expression obtained will logically be the same as for the conventional expression.

This command has three arguments: the progression (the first) and the upper and lower summation limits (second and third, respectively). The summation limits can be whole numbers (including negative numbers) or polynomials with whole number coefficients.

If the user wishes to sum an infinite series, i.e. the sum from a coefficient n to infinity, different functionality must be used in WIRIS: limits, which are explained in the chapter Analysis. The following example demonstrates how these functionalities can be combined.