If the sum of p terms of an AP is equal to sum of its q terms, then prove that the sum of (p+q) terms of it is equal to zero.
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Sp=Sq
p/2[2a +(p-1)d]=q/2 [2a+(q-1)d]
p[2a +pd - d]=q[2a+qd - d ]
2ap + p2d - pd =2aq + q2d -qd
2ap-2aq +p2d -q2d -pd +qd =0
2a (p-q) +(p+q)(p-q)d -d(p-q)=0
(p-q)[2a + (p+q)d - d ]=0
2a + (p+q)d - d=0
2a + [(p+q)-1]d=0
p/2[2a +(p-1)d]=q/2 [2a+(q-1)d]
p[2a +pd - d]=q[2a+qd - d ]
2ap + p2d - pd =2aq + q2d -qd
2ap-2aq +p2d -q2d -pd +qd =0
2a (p-q) +(p+q)(p-q)d -d(p-q)=0
(p-q)[2a + (p+q)d - d ]=0
2a + (p+q)d - d=0
2a + [(p+q)-1]d=0
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