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if the pth qth and rth terms of an ap be a b and c respectively then show that a(q-r)+b(r-p)+c(p-q)=0

 
if the pth qth and rth terms of an ap be a b and c respectively then show that a(q-r)+b(r-p)+c(p-q)=0

Grade:12th pass

1 Answers

Vikas TU
14149 Points
one year ago
Dear student 
Let a = first term of the AP.
and 
Let d = common difference of the AP
Now
a = A+(p-1).d.......(i)
b = A+(q-1).d.......(ii)
c = A+(r-1).d........(iii)
Subtracting 2nd from 1st , 3rd from 2nd and 1st from 3rd we get
a-b = (p-q).d......(iv)
b-c = (q-r).d........(v)
c-a = (r-p).d.......(vi)
multiply iv,v,vi by c,a,b respectively we have
c.(a-b) = c.(p-q).d......(vii)
a.(b-c) = a.(q-r).d........(viii)
b.(c-a) = b.(r-p).d.......(ix)
a(q-r).d+b(r-p).d+c(p-q).d = 0(a(q-r)+b(r-p)+c(p-q)).d = 0
Now since d is common difference it should be non zero
Hence
a(q-r)+b(r-p)+c(p-q)= 0

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