Question
Sun March 20, 2011 By: Waryaam Singh

if the pth, qth and rth term of an AP are x,y and z resp.show that x(q-r)+y(r-p)+z(p-q)=0

Expert Reply
Sat March 19, 2011
Dear Student,
Here is the solution to your problem:-
Let a and d be ist term and common difference of the series AP
Then
x = a+(p-1).d.......(1)
y = a+(q-1).d.......(2)
z = a+(r-1).d........(3)
subtracting 2 from 1, 3 from 2 and 1st from 3rd we get 
x-y = (p-q).d......(4)
y-z = (q-r).d........(5)
z-x = (r-p).d.......(6)
multiply 4,5,6 by z,x,y respectively we have
z.(x-y) = z.(p-q).d......(4)
x.(y-z) = x.(q-r).d........(5)
y.(z-x) = y.(r-p).d.......(6)
x(q-r).d+y(r-p).d+z(p-q).d = 0
(x(q-r)+y(r-p)+z(p-q)).d = 0
Now since d is common difference it should be non zero hence
x(q-r)+y(r-p)+z(p-q)= 0
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