Question

If the pth, qth, rth term of an AP be x, y, z respectively, show that x(q-r)+y(r-p)+z(p-q) = 0

Answer 1

If a is initial term and d is common difference, then from given data,
x = a + (p - 1)d
y = a + (q - 1 )d
z = a + (r - 1 )d
Now taking the given expression, substitute for x, y and z from above.
x(q-r)+y(r-p)+z(p-q) = (a + (p - 1)d)(q - r) + (a + (q - 1)d)(r - p) + (a + (r - 1)d)(p - q)
= a{q - r + r - p + p - q} - d{{q - r + r - p + p - q} + p(q -r) + q(r - p) + r (p - q)
= a (0) - d(0) + pq - pr + qr - pq + rp - rq
= 0 + 0 + 0 = 0 ...Proved.