if the Pth, Qth & Rth term of an A.P are a , b, c, respectively, then show that (a-b)R+(b-c)P+(c-a)Q=0
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a = t1+(P-1)d
b = t1+(Q-1)d
c = t1+(R-1)d
a-b = Pd - Qd
b - c = Qd - RD
c - a = Rd - Pd
(a-b)R + (b-c)P + (c-a)Q
⇒ (Pd - Qd)R + (Qd-Rd)P + (Rd - Pd)Q
⇒ PRd - QRd + PQd - PRd + QRd - PQd
⇒ 0
b = t1+(Q-1)d
c = t1+(R-1)d
a-b = Pd - Qd
b - c = Qd - RD
c - a = Rd - Pd
(a-b)R + (b-c)P + (c-a)Q
⇒ (Pd - Qd)R + (Qd-Rd)P + (Rd - Pd)Q
⇒ PRd - QRd + PQd - PRd + QRd - PQd
⇒ 0
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