If p-th-q-th- and r-th- terms of an A-P- are a-b-c respectively- then the value of-a-b-r-b-c-p-c-a-q

Question

Toppr Admin · Accepted Answer

Let us considerFirst term -ACommon difference -Dthen a-A-p-x2212-1-Db-A-q-x2212-1-Dc-A-r-x2212-1-DGiven-a-x2212-b-r-b-x2212-c-p-c-x2212-a-q-A-p-x2212-1-D-x2212-A-x2212-q-x2212-1-D-r-A-q-x2212-1-D-x2212-A-x2212-r-x2212-1-D-p-A-r-x2212-1-D-x2212-A-x2212-p-x2212-1-D-q-p-x2212-q-Dr-q-x2212-r-Dp-r-x2212-p-Dq-D-pr-x2212-qr-qp-x2212-rp-rq-x2212-pq-D-0-0

If pth,qth and rth terms of an A.P. are a,b,c respectively, then find the value of:(a−b)r+(b−c)p+(c−a)q

Let us considerFirst term =ACommon difference =Dthen a=A+(p−1)Db=A+(q−1)Dc=A+(r−1)DGiven(a−b)r+(b−c)p+(c−a)q=[A+(p−1)D−A−(q−1)D]r+[A+(q−1)D−A−(r−1)D]p+[A+(r−1)D−A−(p−1)D]q=(p−q)Dr+(q−r)Dp+(r−p)Dq=D[pr−qr+qp−rp+rq−pq]=D(0)=0

If $p^{t h}, q^{t h}$ and $r^{t h}$ terms of an A.P. are $a, b, c$ respectively, then find the value of:
$(a - b) r + (b - c) p + (c - a) q$