2016/003) In an AP sum of p terms is same as sum of q terms. Show that $(p+q)^{th}$ term is zero

let $1^{st}$ term be $a$ and difference $d$
let $k^{th}$ term be $t_k$ and sum upto k term be $S_k$
$t_{p}= a + (p - 1) d$
So $S_p= \frac{a + (p-1) d}{2} * p$
sum of q terms $S_q= \frac{a + (q- 1)d}{2}* q = \frac{a + (p-1) d}{2}* p$ (as both are same)
So $(a-d) q + dq ^2 = (a-d) p + dp^2$
or $(a-d)(q-p) = d(p^2 - q^2)$
so $(a-d) = -d (p+ q)$
or $a + (p+q-1) d = 0$ and this is the $(p+q)^{th}$ term and it is zero