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Question
The product of three increasing numbers in GP is 5832. If we add 6 to the second number and 9 to the third number, then resulting numbers form an AP. Find the numbers in GP
Solution
Let the increasing numbers in G.P be, a, ar.
Given `"a"/"r" xx "a" xx "ar"` = 5832
⇒ a3 = 5832 = 183
⇒ a = 18
Also given `"a"/"r"`, a + 6, ar + 9 form an A.P.
∴ 2(a + b) = `"a"/"r" + ("ar" + 9)`
⇒ (a + 6) + (a + 6) = `"a"/"r" + ("ar" + 9)`
⇒ `("a" + 6) - "a"/"r"` = (ar + 9) – (a + 6)
⇒ `"a" + 6 - "a"/"r"` = ar + 9 – a – 6
⇒ `"a" + 6 - "a"/"r"` = ar – a + 3
Substituting the value of a = 18, we get
`18 + 6 - 18/"r"` = 18r – 18 + 3
`24 - 18/"r"` = 18r – 15
24 + 15 = `18"r" + 18/"r"`
39 = `18("r" + 1/"r")`
39 = `(18("r"^2 + 1))/"r"`
39r = 18r2 + 18
18r2 – 39r + 18 = 0
(2r – 3)(3r – 2) = 0
2r – 3 = 0 or 3r – 2 = 0
r = `3/2` orr = `2/3`
Case (i): When a = 18, r = `3/2` the numbers in G.P are
`18/(3/2), 18, 18 xx (3/2)`
⇒ `36/3, 18, 27, 12, 18, 27`
⇒ 27, 18, 12
Case (ii): When a = 18, r = `2/3`
The number in G.P are `18/(2/3), 18, 18 xx 2/3`
⇒ `54/2, 18, 36/3`
⇒ 27, 18, 12
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