Progression

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PROGRESSION. That state of a business which is neither the commencement nor the end. Some act done after the matter has commenced and before it is completed. Plowd. 343. Vide Consummation; Inception.

A Law Dictionary, Adapted to the Constitution and Laws of the United States. By John Bouvier. Published 1856.
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We call the finite geometric progression [a.sub.kt+1], [a.sub.kt+2], [a.sub.kt+3], ..., [a.sub.kt+t] the (k + 1)th period of {[a.sub.n]} and [a.sub.(kt+1)t], [a.sub.(kt+1)t+1] the (k+1)th interval of {[a.sub.n]}, [q.sub.1] is named the common ratio inside the periods and [q.sub.2] is called the interval common ratio, t is called the number sequence {[a.sub.n]}'s period.
The most common use of this is in the arithmetic or geometric progressions. We intend to show that after introducing the concept of sequence and series, a teacher can introduce convolution of these series.
In this paper, we extend the concept to sequences of numbers in geometric progression with alternate common ratios and the periodic sequence with two common ratios.
Chapters discuss the pigeonhole principle, the greatest common divisor, squares, digital sums, arithmetic and geometric progressions, complementary sequences, quadratic functions and equations, parametric solutions for real equations, the scalar project, equilateral triangles in the complex plane, recurrence relations, sequences given by implicit relations, and matrices associated to second order recurrences.
Some of their works deal with simple geometric progressions, but the most interesting pieces are the two video installations.
Passmore discusses Polya's "worthwhile and interesting problems" while Berenson presents arithmetic and geometric progressions as seen in arrays of numbers.
The geometric progressions of the choreography resonated in the design of a silver bridge that bisected the back of the stage.
He expanded this concept into geometric progressions as the figures were either diminished or expanded to create a new division of the plane.
He discusses continued fractions; the geometry of complex numbers, quaternions, and spins; and Euler groups and the arithmetic of geometric progressions. He ends with 79 problems for students ages five to 15, with solutions.
In our previous encounter with the Figure 8 array, we were not in a position to benefit from the fact that the first row and the first column were both geometric progressions: (1, 2, 4, 8, 16, 32 and 1, 3, 9, 27, 81, 243).