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If x,y and z are pth ,qth and rht terms, respectively of an A.P and also of a G.P then prove that `x^(x-y)y^(z-x)z^(x-y)=1`

Text Solution

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Let a, d be the first term and common difference of an AP and b, r be the first term and common ratio of a GP.
Then, `x = a + (m -1) d and x = br^(m-1)`
`y = a + (n -1) d and y = br^(n-1)`
`z = a + (p -1)d and z = br^(p-1)`
Now, `x - y = (m -n) d, y - z = (n -p) d`
and `z -x = (p -m) d`
Again now, `x^(y-2).y^(z-x).z^(x -y)`
`= [br^(m-1)]^((n-p)d).[br^(n-1)]^((p -m)d).[br^(p-1)]^((m-n)d)`
`= b^([n -p + p -m + m -n]d).r^([(m -1) (n-p) + (n-1) (p-m) + (p -1) (m-n)]d)`
`= b^(0) .r^(0) = 1`
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