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Question
If three quantities are in continued proportion; show that the ratio of the first to the third is the duplicate ratio of the first to the second.
Solution
Let x, y and z be the three quantities which are in continued proportion.
Then, x : y :: y : z ⇒ y2 = xz ...(1)
Now, we have to prove that
x : z = x2 : y2
That is we need to prove that
xy2 = x2z
LHS = xy2 = x(xz) = x2z = RHS ...[Using (1)]
Hence, proved.
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