Arithmetic Sequences

An arithmetic sequence is a sequence in which the difference between each consecutive term is constant. An arithmetic sequence can be defined by an explicit formula in which a _n = d (n - 1) + c , where d is the common difference between consecutive terms, and c = a ₁ . An arithmetic sequence can also be defined recursively by the formulas a ₁ = c, a _n+1 = a _n + d , in which d is again the common difference between consecutive terms, and c is a constant.

The sum of an infinite arithmetic sequence is either ∞ , if d > 0 , or - ∞ , if d < 0 .

There are two ways to find the sum of a finite arithmetic sequence. To use the first method, you must know the value of the first term a ₁ and the value of the last term a _n . Then, the sum of the first n terms of the arithmetic sequence is S _n = n() . To use the second method, you must know the value of the first term a ₁ and the common difference d . Then, the sum of the first n terms of an arithmetic sequence is S _n = na ₁ + (dn - d ) .