Sum of n terms of an Arithmetic Progression

Sum of n terms of an Arithmetic Progression
1.Theorem : Let a be the first term, d be the common difference, n be the number of terms and s_n be the sum of first n terms of an Arithmetic progression , then
\(\boxed{{S_n} = \frac{n}{2}[2a + (n - 1)d}\)
Proof :    Let \(a,\,a + d,\,a + 2d,...... + a + (n - 1)d\) be n terms of an arithmetic progression.
 We have 
 \({S_n} = a + (a + d) + (0 + 2d) + ....... + (a + n - 1)d\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \to (1)\)
\({S_n} = [a + (n - 1)d] + [a + (n - 2)d] + [a + (n - 3)d] + .........\, + (a + 2d) + (a + d) + a\,\,\,\,\,\,\,\,\,\, \to (2)\)
Adding the above two equations, we get
2S_{n=\(\left[ {2a + (n - 1)d} \right] + \left[ {2a + (n - 1)d} \right] + .......... + \left[ {2a + (n - 1)d} \right]\)
\(\Rightarrow 2{S_n} = n\left[ {2a + (n - 1)d} \right]\)
\(\Rightarrow {S_n} = \frac{n}{2}\left[ {2a + (n - 1)d} \right]\)}
2. Theorem . If 'a' is the first term and '\(l\)' is the last term, then sum to n terms of an A.P \({S_n} = \frac{n}{2}\left[ {a + l} \right]\)
Proof :We have \({S_n} = \frac{n}{2}\left[ {2a + (n - 1)d} \right]\)
\(\Rightarrow {S_n} = \frac{n}{2}\left[ {a + a + (n - 1)d} \right]\)
\(\Rightarrow {S_n} = \frac{n}{2}\left[ {a + l} \right]\)
3.     If S_m =n and S_n =m for an A.P then S_m+n = -(m+n)
4.    If the sum of n terms of a sequence is Sn =An²+Bn+C (i.e Quadratic expressionin n), then the sequence is A.P, with first term is 3A+B and common difference is 2A. Also in this sequence nth term T_n=2An+(A+B).
5.    If the ratio of the sums of n terms of two A.P's is given then the ratio of their nth terms may be obtained by replacing n with (2n-1) in the given ratio
6.    If the ratio of n^th terms of two A.P's is given then the ratio of the sums of their n terms may be obtained by replacing n with \(n+1 \over 2\) in the given  ratio
7.    Sum of the interior angles of a polygon of 'n' sides is (n-2)180⁰
8.    The nth common terms of two Arithmetic series is (L.C.M of common difference of 1st series and 2nd series) (n-1)+1^st common term of both series.
9.    If the sn of n terms of a sequence is given, then n^th term a_n of the sequence can be determined by using the following \({a_n} = {s_n} - {s_{n - 1}}\)