if the sum of first p terms of an a. p. is equal to the sum of first q terms then show that the sum of its first ( p+q) terms is zero. (p is not equal to zero)
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sumof pterms in ap=p/2(2a+(p-1)d)
sumof qterms inap=q/2(2a+(q-1)d)
said both are equal
p/2(2a+(p-1)d)=q/2(2a+(q-1)d)
2ap+p(p-1)d=2aq+q(q-1)d
2a(p-q)+(p-q)(p+q-1)d=0
2a+(p+q-1)d=0
p+q/2(2a +(p+q-1)d)=0
hence the sum of its first ( p+q) terms is zero
sumof qterms inap=q/2(2a+(q-1)d)
said both are equal
p/2(2a+(p-1)d)=q/2(2a+(q-1)d)
2ap+p(p-1)d=2aq+q(q-1)d
2a(p-q)+(p-q)(p+q-1)d=0
2a+(p+q-1)d=0
p+q/2(2a +(p+q-1)d)=0
hence the sum of its first ( p+q) terms is zero
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