If the first, second and the last terms of an A.P. are a, b, c respectively, then the sum is

We have, first term = a, ∴T1 = aSecond term = b, ∴T2 = bThen common difference d = T2 – T1 = b – aAlso, last term = c.⇒ c=a+(n−1)d⇒n=c−a+dd.⇒ n=(b+c−2a)(b−a)       (∵d=b−a)∴ Sum of n terms Sn=n2(a+l)=(b+c−2a)(a+c)2(b−a)