Arithmetic Progression.

Arithmetic Progression. Arithmetic Progression, sequence of numbers that increase or diminish by a common difference so that any number in the sequence is the arithmetic mean, or average, of the numbers preceding and following it (See also Geometric Progression; Sequence and Series). The numbers 7, 10, 13, 16, 19, 22 form an arithmetic progression, as do the numbers 12, 10y, 9, 7y, 6. The natural numbers 1, 2, 3, 4 form an arithmetic progression in which the difference is 1. To find the sum of any arithmetic progression, multiply the sum of the first and last terms by half the number of terms. Thus, the sum of the first ten natural numbers is (1 + 10) × (10 ÷ 2) = 55. The general arithmetic progression is a, a + d, a + 2d, a + 3d,... where the first term, a, and the common difference, d, are arbitrary numbers. The n th term of this progression (often denoted by the term an) is given by the formula an = a + (n - 1) d, and the sum of the first n terms is yn [2a + (n - 1)d ], or yn (a + an). Microsoft ® Encarta ® 2009. © 1993-2008 Microsoft Corporation. All rights reserved.