If sum of n term is m and sum of m term is n than prove that sum of m+n term is equal to -(m+n)

If sum of n term is m and sum of m term is n than prove that sum of m+

1.	If sum of n term is m and sum of m term is n than prove that sum of m+n term is equal to -(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)AndS n= 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 = 2 n - 2 m2a (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)Hence Proved.\xa0