Solve this:
Q28. If pth , qth and rth term of an A.P. are a, b, c respectively, then show that
a(q – r) + b(r – p) + c (p – q) = 0.
↵Dear Student,
Regards
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 (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
Regards