If #a,b and c# are the #p^(th)#, #q^(th)# and #r^(th)# term of an AP then show that #p(b-c)+q(c-a)+q(a-b)=0#?

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Answer 1 · 2018-05-31T17:22:51

See below

Note there is a typo in the question:

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

The AP, with common difference #delta# and first term #alpha#, is:

#alpha + (alpha + delta) + ..... + underbrace( (alpha + (p-1) delta))_(a = "p-th term")+ ..... + underbrace((alpha + (q-1) delta ))\_(b = "q-th term")+ ....... + underbrace((alpha + (r-1) delta))\_(c = "r-th term") + ......#

#bb implies p(b-c)+q(c-a)+ r (a-b)#

#= p(alpha + (q-1) delta- alpha - (r-1) delta)+q(alpha + (r-1) delta - alpha - (p-1) delta)+ r (alpha + (p-1) delta -alpha - (q-1) delta)#

Eliminate #alpha#'s:

#= p( (q-1) delta- (r-1) delta)+q( (r-1) delta - (p-1) delta)+ r ( (p-1) delta - (q-1) delta) #

Factor out #delta#:

#= delta( p( (q-1)- (r-1))+q( (r-1) - (p-1))+ r ( (p-1) - (q-1)) )#

Expand:

#= delta( p( q- r)+q( r - p )+ r ( p - q) )#

#= delta( color(red)(pq)- color(blue)(pr) + bb(q r )- color(red)(q p) + color(blue)(r p) - bb(r q) ) bb color(green)(= 0)#