If \(a^{th}\), \(h^{th}\), \(r^{th}\) terms of an \(A.P.\) are \(x\), \(y\), \(z\) respectively,
then verify that \(x(h - r) + y(r - a) + z(a - h) = 0\).
Proof:
Let \(a\) be the first term and \(d\) be the common difference of an \(A.P.\)
\(a^{th} term = x\), That is \(t_a = x\)
\(h^{th} term = y\), That is \(t_h = y\)
\(r^{th} term = z\), That is \(t_r = z\)
Using the general formula:
\(t_a = a + (a - 1)d = x\) - - - - (1)
\(t_h = a + (h - 1)d = y\) - - - - (2)
\(t_r = a + (r - 1)d = z\) - - - - (3)
Now, using the above equations, we get:
\(x(h - r) + y(r - a) + z(a - h)\)
\(= [a + (a - 1)d](h - r) + [a + (h - 1)d](r - a) + [a + (r - 1)d](a - h)\)
\(= [a(h - r) + d(a - 1)(h - r)] + [a(r - a) + d(h - 1)(r - a)] + [a(a - h) + d(r - 1)(a - h)]\)
\(= a[(h - r) + (r - a) + (a - h)] + d[(a - 1)(h - r) + (h - 1)(r - a) + (r - 1)(a - h)]\)
\(= a[h - r + r - a + a - h] + d[ah - ar - h + r + hr - ha - r + a+ ra - rh - a + h]\)
\(= a(0) + d(0)\)
\(= 0\)
Thus, \(x(h - r) + y(r - a) + z(a - h) = 0\).
Hence, we proved.