Question

if the sum of the first m term of an ap in n and the sum of first n term is m. then show that the sum of its first m+n terms is - m+n

Answer 1

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)