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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.

**Explanation:**

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

⇒ a^{2 }- d^{2} = 5d

⇒ 6^{2 }- d^{2} = 5d

⇒ 36^{ }- d^{2} = 5d

⇒ d^{2} + 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.

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