There are three numbers in geometric progression. Their product is 1728 and the sum of the products of the numbers taken two at a time is 504. Find the largest of the three numbers.
Explanation:
Let the three terms in GP be a/r, a and ar.
∴ a/r × a × ar = 1728 ⇒ a3 = 1728 ⇒ a = 12
Also, a/r × a + a × ar + ar × a/r = 504 ⇒ a2/r + a2 + a2r = 504 ⇒ 2/r + 2 + 2r = 7 [∵ a = 12] ⇒ 2r2 – 5r + 2 = 0 ⇒ (2r - 1)(r - 2) = 0 ⇒ r = ½ or 2
Case 1: r = ½ Largest number will be a/r = 24.
Case 1: r = ½ Largest number will be ar = 24.
In any case largest number will be 24.
Hence, option (a).
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