Show that x(q-r) + y(r-p) + z(p-q) = 0, Mathematics

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If the p^th, q^th & r^th term of an AP is x, y and z respectively, show that x(q-r) + y(r-p) + z(p-q) = 0

Ans: p^th term ⇒ x = A + (p-1) D

q^th term ⇒ y = A + (q-1) D

r^th term ⇒ z = A + (r-1) D

T.P x(q-r) + y(r-p) + z(p-q) = 0

={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 {(p-1)(q-r)

+ (r-1) (r-p) + (r-1) (p-q)}

⇒ A.0 + D{p(q-r) + q(r-p) + r (p-q)

- (q-r) - (r-p)-(p-q)}

= A.0 + D.0 = 0.

Hence proved

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