Question

If the p th, q th and r th term of an A.P. are a, b, and c respectively, then show that : -

(a - b)r + (b - c)p + (c - a)q = 0

Answer 1

Heya!!

Let X be first no of an AP and D be common difference.

Fr ur N :p

Answer 2

${\huge{\bold {Solution :-}}}$

X + ( p-1) D = a - - - - - - - (1)
X + ( q-1) D = b - - - - - - - (2)
X + (r-1 ) D = c - - - - - - - (3)

${\underline {\huge {\bold {\boxed {To \: Show}}}}}$
(a-b) r + (b-c) p + (c-a) q=0

Substitute equation 2 by 1, 1 by 3, 3 by 2.

➡️ x + (p-1)D - x + (q-1)D = a-b
➡️ pD - qD + D + D = a - b
➡️ D (p-q) = a-b-------------------(4)

${\huge {\bold {\mathfrak{Similarly :-}}}}$

➡️ D(q - r) = b - c (5)
➡️ D ( r - p) = c - a (6)

Put values of RHS
➡️ d( p - q) r + D(q - r) p + D( r - p) q
➡️ Dpr - Dqr + Dpq - Drp + Drq - Dpq
➡️ 0

➡️ ➡️ ➡️ ➡️ ➡️ ➡️ ➡️ ➡️ ➡️ ➡️ ➡️ ➡️ ➡️ ➡️ ➡️ ➡️ ➡️

Proved :- (a-b) r+(b-c) p + (c-a) q=0

➡️ ➡️ ➡️ ➡️ ➡️ ➡️ ➡️ ➡️ ➡️ ➡️ ➡️ ➡️ ➡️ ➡️ ➡️ ➡️ ➡️

HOPE IT WILL HELP YOU

➡️➡️➡️➡️➡️➡️➡️➡️➡️➡️➡️➡️➡️➡️➡️➡️➡️➡️➡️➡️➡️➡️

${\huge{\bold{\mathfrak{Be \: Brainly}}}}$

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