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The product of three numbers in G.P. is 125 and the sum of their products taken in pairs is 87 1/2. Find them.

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Let the three numbers be a/r, a, ar

So, according to the question

a/r × a × ar = 125 … equation (1)

From equation (1) we get,

a3 = 125

a = 5

a/r × a + a × ar + ar × a/r = 87 1/2

a/r × a + a × ar + ar × a/r = 195/2

a2/r + a2r + a2 = 195/2

a2 (1/r + r + 1) = 195/2

Substituting a = 5 in above equation we get,

52 [(1 + r+ r)/r] = 195/2

1 + r+ r = (195r/2 × 25)

2(1 + r+ r) = 39r/5

10 + 10r2 + 10r = 39r

10r2 – 29r + 10 = 0

10r2 – 25r – 4r + 10 = 0

5r(2r - 5) – 2(2r  -5) = 0

r = 5/2, 2/5

So G.P is 10, 5, 5/2 or 5/2, 5, 10

∴ The three numbers are 10, 5, 5/2 or 5/2, 5, 10

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