Question

if three quantities are such that the first to the third is the duplicate ratio of the first to the second , prove that they are in continued proportion

Answer 1

Let numbers are a, b, and c

According to the given information,

$\frac{a}{c} = \frac{ {a}^{2} }{ {b}^{2} } \\ \\ \frac{1}{c} = \frac{a}{ {b}^{2} } \\ \\ a = \frac{ {b}^{2} }{c} - - - - - - 1equation \\ \\ \\ = = = = = = = = = = = = = = = = = = = =$

Suppose a, b and c are in continued proportion,


So,

a : b should equal to b : c

Then,

Putting the value of a from 1equation


$\frac{ \frac{ {b}^{2} }{c} }{b} \: \: \: \: should \: equal \: to \: \: \: \frac{b}{c} \\ \\ \frac{ {b}^{2} }{c} \times \frac{1}{b} \: \: \: should \: equal \: to \: \frac{b}{c} \\ \\ now \\ \\ \\ \frac{b}{c} = \frac{b}{c}$



So, our supposing was correct.


Hence, a, b and c are in continued proportion




I hope this will help you

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Answer 2

Answer:hope it helps u

Step-by-step explanation:i have explained  every point