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Question
If x, 2y, 3z are in A.P., where the distinct numbers x, y, z are in G.P. then the common ratio of the G.P. is ______.
Options
3
`1/3`
2
`1/2`
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Solution
If x, 2y, 3z are in A.P., where the distinct numbers x, y, z are in G.P. then the common ratio of the G.P. is `1/3`.
Explanation:
Since x, 2y, 3z are in A.P
∴ 2y – x = 3z –2y
⇒ 4y = x + 3z .....(i)
Now x, y, z are in G.P.
∴ Common ratio r = `y/x = z/y` ....(ii)
∴ y2 = xz
Putting the value of x from equation (i), we get
y2 = (4y – 3z)z
⇒ y2 = 4yz – 3z2
⇒ 3z2 – 4yz + y2 = 0
⇒ 3z2 – 3yz – yz + y2 = 0
⇒ 3z(z – y) – y(z – y) = 0
⇒ (3z – y)(z – y) = 0
⇒ 3z – y = 0 and z – y = 0
⇒ 3z = y and z ≠ y ....[∵ z and y are distinct numbers]
⇒ `z/y = 1/3`
⇒ r = `1/3` ...[From equation (ii)]
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