arithmetic series

An arithmetic series is the series, $\sum_{i=1}^na_i$ , 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: $$S_n=\sum_{i=1}^na_i =a_1+a_1+d+\cdots +a_1+(n-2)d+a_1+(n-1)d$$ 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 \begin{eqnarray} 2S_n=n(a_1+a_n)\\ S_n= \frac{1}{2}n(a_1+a_n) \end{eqnarray}Hence, by substituting $a_n=a_1+(n-1)d$ we get the first formula.