Question

in an AP the Pth is a the a term is qth term is b rth term is c prove that a( p-q ) b (q- r) c (r- p)is 0​

Answer 1

Step-by-step explanation:

Let A be the first term and D the common difference of A.P.

$T_{p}$ = a = A + (p-1)D = (A-D) + pD .......... equation 1

$T_{q}$ = b = A + (q-1)D = (A-D) + qD .......... equation 2

$T_{r}$ = c = A + (r-1)D = (A-D) + rD    .......... equation 3

Here we have got two unknowns A and D which are to be eliminated.

we multiply equation 1,2 and 3 by (p-q) , (q-r) and (r-p) respectively and add all of them:

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

= (A-D)(p - q + q - r + r - p) + D( r(p-q) + p(q-r) + q(r-p) )

= (A-D)(0) + D(pr - qr + pq - pr + qr - pq)

= 0 + D(0)

=0