I was studying arithmetic mean and read Sum of 'n' arithmetic mean is n*(a+b)/2. But i also read if there is an A.P. with n terms then the sum of those n terms is n times (the middle term if n is odd)-(i) or else if n is even the n times (arithmetic mean of the middle terms).(ii)
can anyone explain how i and ii have been concluded.
Hint:
Prove that, if $a_1, a_2, \dots, a_n$ is an arithmetic progression, for any k such that $1\le k\le n$, $a_k+a_{n-k+1}$ is constant.
From this result you can deduce the high-school formula $$a_1 + a_2+\dots+ a_n =n\,\frac{a_1+a_n}2.$$ Now if $n$ is odd: $n=2m+1$, it happens that $\dfrac{a_1+a_n}2$ is the middle term $a_{m+1}$, and if $n=2m$, $\;\dfrac{a_1+a_n}2=\dfrac{a_m+a_{m+1}}2$.