**Solution:** we have given

S_{p}=S_{q}

p/2[2a +(p-1)d]=q/2 [2a+(q-1)d]

p[2a +pd - d]=q[2a+qd - d ]

2ap + p^{2}d - pd =2aq + q^{2}d -qd

2ap-2aq +p^{2}d -q^{2}d -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

Multiply both sides with (p + q)/2