Plz Solve this and Explain also
Dear Student,
Let A and D be the first term and common difference of A.P.
ap = a ⇒A + (p– 1) D = a … (1)
aq = b ⇒A + (q – 1) D = b … (2)
ar = c ⇒ A + (r – 1) D = c … (3)
a)
. a (q – r) + b (r – p) + c (p – q)
. a (q – r) + b (r – p) + c (p – q)
= [A + (p– 1) D] (q – r) + [A + (q – 1) D] (r – p) + [A + (r – 1) D] (p – q)]
= A (q – r) + (p – 1) (q – r)D + A (r – p) + (q – 1) (r – p) D + A (p – q) + (r – 1) (p – q) D
= A × (q – r + r – p + p – q) + D × (pq – pr – q + r + qr – pq – r + p + pr – rq – p + q)
= A × 0 + D × 0
= 0
b)
a (q – r) + b (r – p) + c (p – q) = 0 (Proved)
a (q – r) + b (r – p) + c (p – q) = 0 (Proved)
⇒ aq – ar + br – bp + cp – cq = 0
⇒ r (–a + b) + p (–b + c) + q (–c + a) = 0
⇒ – r (a – b) – p (b – c) – q (c – a) = 0
⇒ r (a – b) – p(b – c) + q (c – a) = 0
Regards!