Question

If the sum of m term of A.P. is same as the sum of n term show that the sum of its m+n term is zero.

Answer 1

Sn=(n/2)(2a+(n-1)d) ....eq(i)
Sm=(m/2)(2a+(m-1)d) ....eq(ii)
Sm+n=((m+n)/2)(2a+(m+n-1)d) .....eq(iii)
given, Sn=Sm
therefore,
2am+dm^2-md = 2an+dn^2-nd
2a(m-n)+d(m^2-n^2-(m-n))=0
m-n(2a+d(m+n-1)=0
2a+d(m+n-1)=0 .... eq(iv)
from eq(iii) and eq(iv) we get
Sm+n=0

Hope its help you
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