arithmetic series

An arithmetic series is the series, $\\sum_{i=1}^{n}a_{i}$ , in which each real term has the form $a_{i}=a_{i-1}+d$ for $i=2,\\ldots,n$ where $d$ is constant. The sum of the sequence is given by the following $\\displaystyle\\frac{1}{2}n[2a_{1}+d(n-1)].$ In order to find the formula above firstly we express the terms of the sequence, $a_{2},\\ldots,a_{n}$ in terms of $a_{1}$ and the constant $d$ . In this case we get $a_{2}=a_{1}+d,a_{3}=a_{2}+2d,\\ldots,a_{n}=a_{1}+(n-1)d$ . Now we express the sum of the sequence by developing the series forward and we have:

S_{n}=\\sum_{i=1}^{n}a_{i}=a_{1}+a_{1}+d+\\cdots+a_{1}+(n-2)d+a_{1}+(n-1)d

Reversely, we develop the series backwards and we get

S_{n}=a_{n}-d+a_{n}-2d+\\cdots+a_{n}-(n-1)d

It is easily seen that by adding the two expressions we get

	$\\displaystyle 2S_{n}=n(a_{1}+a_{n})$		(1)
	$\\displaystyle S_{n}=\\frac{1}{2}n(a_{1}+a_{n})$		(2)

Hence, by substituting $a_{n}=a_{1}+(n-1)d$ we get the first formula.