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# Def/Arithmetic progression

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Definition of Arithmetic progression: Suppose that \$(a_i)\$ is a sequence, indexed by an interval of integers. The values of \$a_i\$ can be integers, real numbers, or most generally, elements of an abelian group.

We say that the sequence \$(a_i)\$ is an arithmetic progression, or linear progression, if there exists a constant (integer, real number, or element of the appropriate abelian group) \$m\$ such that:

• The sequence of differences \$(a_i')\$ is a constant sequence, with constant \$m\$.
• Equivalently, for every nonmaximal \$i\$ in the indexing set, \$a_{i+1} = a_i + m\$.

In this case, we call \$m\$ the slope or step size of the arithmetic progression \$(a_i)\$.

## Logical Connections

This definition logically relies on the following definitions and statements: Def/Sequence, Def/Sequence of differences

The following statements and definitions logically rely on the material of this page: Def/Quadratic sequence, and State/Arithmetic progression rule for binary quadratic forms

To visualize the logical connections between this definition and other items of mathematical knowledge, you can visit any of the following clusters, and click the "Visualize" tab: Clust/Sequences

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