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If the pth , qth and rth term of an AP is x, y and z Then show that x( q-r) y( r-p) z( p-q) = 0?
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If the pth , qth and rth term of an AP is x, y and z Then show that x(...
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If the pth , qth and rth term of an AP is x, y and z Then show that x(...
Let A be the first term
d be the difference

then pth term is = A+(p-1)d
x=A+(p-1)d. ..... eq(1)
similarly
Y= A+(q-1)d,.........eq(2)
z=A+(r-1)d................eq(3)

now on subtract
x-y=(p-q)d.......eq( 4)
y-z=(q-r)d..........eq(5)
z-x=(r-p)d............eq(6)

so now multiply eq 1,2,3 respectively by
z,x,y
then z(x-y)=z(p-q)d........eq(7)
x(y-z)=x(q-r)d...............eq(8)
y(z-x)=y(r-p)d......,.........eq(9)
from eq 7, 8, 9 on addition we get

so x (q-r)+ y(r-p)+z (p-q)=0
proved.
(Q. E. D)

That's all🙂
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