Question

If $x, y, z$ are $p^{ ext{th}}, q^{ ext{th}}, r^{ ext{th}}$ terms respectively of an A.P., then prove that $p(y - z) + q(z - x) + r(x - y) = 0$.

Accepted Answer

Let first term = $a$, common difference = $d$. $x = a + (p-1)d$, $y = a + (q-1)d$, $z = a + (r-1)d$. $y - z = (q - r)d$, $z - x = (r - p)d$, $x - y = (p - q)d$. Substitute into LHS: $p(q-r)d + q(r-p)d + r(p-q)d = d[pq - pr + qr - pq + pr - qr] = 0$.

AP proof for terms p, q, r

Explanation

Practise this under board exam conditions