Mathematical progressions are an integral part of any high school algebra curriculum, defined as any series of numbers that follow a pattern. Two common types of mathematical progressions taught in school are geometric progressions and arithmetic progressions. Different properties of arithmetic progressions can be incorporated into school projects.

## Defintion

An arithmetic progression is any series of numbers in which each term has a constant difference with the preceding term. For example, "1,2,3..." is an arithmetic progression, because each term is one greater than the one preceding. To teach this to students, have them create arithmetic progressions given a common difference. Another activity is to have them identify which progressions are arithmetic and find the common difference between the terms.

## Recursive Formula

The most basic type of formula for any arithmetic progression is the recursive formula. In the recursive formula, a first term is specified as zero (0). The formula is "a(n+1) = a(n) + r," in which "r" is the common difference between subsequent terms. Basic projects that use the recursive formula include constructing the progression from a formula and constructing the formula from an arithmetic progression. This can be an expansion of the project from the previous section.

## Explicit Formula

The explicit formula for an arithmetic progression has the form "a(n) = a(1) + n*r," in which "a(n)" is the nth term (defined as any term in the arithmetic sequence) of the progression, "a(1)" is the first term, and "r" is the common difference. This formula can be easily changed into the recursive form and vice-versa. Have students practice constructing the explicit formula on the recursive formulas they obtained in the Section 2 project.

## Summation

To find the sum of an arithmetic sequence from "a(1)" to "a(n)" with common difference "r," plug the following into the formula: "n(n+1)/2 + r(n)(n-1)/2 + (a(1)-1)*n." Have students use the formula to sum the series of consecutive terms of an arithmetic progression and check their answer with the sum obtained just by adding the terms. Have them compile this with the other activities in Sections 1 to 3 to create their very own project on arithmetic progressions.