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.

Answer: The required numbers are 15, 6, -3 or 2, 6, 10.

Let us proceed step by step to find the answer.

Explanation:

Let us consider the first three numbers of an A.P as (a - d), a, and (a + d) where a is the first term and d is the common difference of AP.

We are taking the sequence as shown above so that while adding the terms, d gets canceled and we can find the value of an easily.

Sum of first three numbers is (a - d) + a + (a + d) = 18 [from the given data]

(a - d) + a + (a + d) = 18

a - d + a + a + d = 18

3a = 18

a = 6 --------(1)

Also, the product of first and third term = 5 times common difference [from the given data]

So, we can write the given equation as (a - d) × (a + d) = 5d

a² - d² = 5d  [on multiplying both the terms in left hand side]

On substituting the value a obtained from equation 1 we get,

⇒ 36 - d² = 5d

⇒ d² + 5d - 36 = 0 [rearranging terms]

⇒ d² + 9d - 4d - 36=0 [splitting middle term]

⇒ d (d + 9) - 4(d + 9) = 0

⇒ (d + 9) × (d - 4) = 0

⇒ d = -9 and d = 4 [equating both the products separately with 0]

If d = -9, the numbers are 15, 6, -3.

If d = 4, the numbers are 2, 6, 10.

Hence, the required numbers are 15, 6, -3 or 2, 6, 10.