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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

(a) 3

(b) 13

(c) 2

(d) 12

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Solution

Let x, 2y, 3z be in A.P, for distinct x, y and z
where x, y, z are in G.P
Since x, 2y and 3z are in A.P
2y=x+3y2 middle term=arithmetic mean of adjacent termsi.e 4y=x+3y ...1
and since x, y and z are in G.P
∴ y2 = xz ...(2)
Let r denote the common ratio of G.P
yx=r and zx=r2 ...3
Now, dividing (1) by x, we get
4yx=1+3yxi.e 4r=1+3r2i.e 3r2-4r+1=0 i.e 3r2-3r-r+1=0i.e 3r-1 r-1=0i.e r=13 or r=1
(r = 1 is not possible since x, y, z are distinct)
Common ratio is 13
Hence, the correct answer is option B.

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