() the total atfiounlt savcu Dj50. If the sum of first m terms of an A.P. is n and sum of first n terms of the same A.P. ism.Show that sum of first (m + n) terms of it is(m+ n).

() the total atfiounlt savcu Dj50. If the sum of first m terms of an A

1.	() the total atfiounlt savcu Dj50. If the sum of first m terms of an A.P. is n and sum of first n terms of the same A.P. ism.Show that sum of first (m + n) terms of it is(m+ n).
Answer» Let a be the first term and d be c.d. of the A P .ThenSm=nn= m/2{2a+ (m-1)d} 2n= 2am+ m( m-1)d. ........(1)andSn= mm= n/2{2a+(n-1)d}2m= 2an+ n(n-1)d. ...........(2)Subtracting eq.(2)- (1), we get2a(m-1)+{m(m-1)- n(n-1)}d = 2n-2m2a(m-n) +{(m^2-n^2)-(m-n)}d = -2(m-n)2a +(m+n-1) d = -2. [On dividing both sides by ( m-n)]………(3) Now,Sm+n=m+n/2{2a +(m+n-1)d}Sm+n=m+n/2(-2) ………[using (3)]Sm+n=-(m+n)