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if the pth , qth and rth terms of an A.P be x,y,z respectively , show that : x(q-r)+y(r-p)+z(p-q)=0.

if the pth , qth and rth terms of an A.P be x,y,z respectively , show that :
x(q-r)+y(r-p)+z(p-q)=0.

Grade:10

1 Answers

Arun
25750 Points
5 years ago

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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