Question

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≠q)​

Answer 1

Step-by-step explanation:

S

p

=S

q

⇒

2

p

(2a+(p−1)d)=

2

q

(2a+(q−1)d)

⇒ p(2a+(p−1)d)=q(2a+(q−1)d)

⇒ 2ap+p

2

d−pd=2aq+q

2

d−qd

⇒ 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

⇒ S

p+q

=0