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. - Mathematics

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.

Sum

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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