Question

x,y,z are three numbers are in A.P and x,y-x ,z-x are in gp show that x/1=y/3=z/5

Answer 1

Answer:

your solution is as follows

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Step-by-step explanation:

$to \: show \: that : \\ \frac{x}{1} = \frac{y}{3} = \frac{y}{5} \\ \\ let \: then \: here \\ \frac{x}{1} = \frac{y}{3} = \frac{z}{5} = k \\ \\ x = k \: , \: y = 3k \: , \: z = 5k \\ \\ now \: here \\ x \: , \: y \: , \: z \: are \: in \: AP \\ y - x = z - y \\ 3k - k = 5k - 3k \\ 2k = 2k \\2 = 2 \: \: \: \: \: \: (1) \\ thus \: verified \\ \\ similarly \: as \: here \\ x \: , \: y - x \: , \: z - x \: are \: in \: GP \\ \\ \frac{y - x}{x} = \frac{z - x}{y - x} \\ \\ \frac{3k - k}{k} = \frac{5k - k}{3k -k } \\ \\ \frac{2k}{k} = \frac{4k}{2k} \\ \\ 2 = 2 \: \: \: \: \: \: (2) \\ \\ since \: (1) = (2) \\ the \: proven \: part \: is \: true \: \\ and \: thus \: proved$