Question

If the sum of the first p terms of an AP is the same as the sum of its first q terms (where p is not equal to q) then show that the sum of its first (p+q) terms is zero

Answer 1

we known according to Ap sum formula
Sn=n/2 {2a+(n-1) d}
now
Sp=p/2 {2a+(p-1) d}-------(1)
sq=q/2 {2a+(q-1) d} --------(2)
a/c
Sp=Sq
p/2 {2a+(p-1) d}=q/2 {2a+(q-1) d}

2pa+pd (p-1)=2aq+qd (q-1)

2a (p-q)+d (p^2-p-q^2+q)

2a (p-q)+d{(p-q)(p+q-1)}=0

(p-q){2a+(p+q-1) d=0

(2a+(p+q-1) d)=0-----------(3)
now
S (p+q)=(p+q)/2 {2a+(n-1) d}
equation (3) put in sum of p+q
S (p+q)=0