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The sum of 3 numbers in AP is 18. If the product of the first and third number is 5 times the common difference, find the numbers.
Let us find the numbers using arithmetic progression properties.
Answer: The numbers are 2, 6, and 10.
Let us use the concept of the general term of AP to find the required numbers.
Let a be the first term of AP and d be the common difference of given AP.
Let the terms of AP are a - d, a, a + d
Given that, the sum of the numbers is 18.
⇒ (a - d) + a + (a + d) = 18
⇒ a - d + a + a + d = 18
⇒ a + a + a = 18
⇒ 3a = 18
⇒ a = 18/3 = 6
Therefore, a = 6
Let us now consider the second information: the product of the first and third number is 5 times the common difference.
So, (a - d) × (a + d) = 5d
⇒ a2 - d2 = 5d
⇒ 62 - d2 = 5d
⇒ 36 - d2 = 5d
⇒ d2 + 5d - 36 = 0
⇒ (d + 9)(d - 4) = 0
Therefore, d = 4
As the terms of AP are a - d, a, a + d, we get their values as 6 - 4, 4, 6 + 4.
⇒ 2, 4, 10 are the terms of given AP.
Therefore, the numbers are 2, 6, and 10 that satisfy the given conditions.