Arithmetic series
In mathematics, an arithmetic progression is a sequence of numbers such that the difference of any two successive members of the sequence is a constant. For instance, the sequence 3, 5, 7, 9, 11, ... is an arithmetic progression with common difference 2.
If the initial term of an arithmetic progression is a and the common difference of successive members is d, then the n-th term of the sequence is given by
- a + nd, n = 0, 1, 2, ... if the initial term is taken as the 0th
- a + (n − 1)d, n = 1, 2, ... if the initial term is taken as the 1st
The first option gives an easier formula, but uses a somewhat confusing terminology.
The sum of the numbers in (an initial segment of) an arithmetic progression is sometimes called an arithmetic series. A convenient formula for arithmetic series is available. The sum S of the first n values of a finite sequence is given by the formula:
- S = ½n(a1 + an)
where a1 is the first term and an the last.
For example to find the sum of the first n positive integers:
- <math>1 + 2 + \cdots + n = \frac{n(n+1)}{2}<math>
known also as the triangular number.
An often-told story is that Gauss discovered this formula when his third grade teacher asked the class to find the sum of the first 100 numbers, and instantly computed the answer, 5050.
See also
