Question

1. The sum of the first three numbers in an AP is 18. If the product of the first and the third term is 5 times the common difference, find the three numbers.​

Answer 1

Step-by-step explanation:

Given:-

• The sum of first three numbers in an AP is 18.

• The product of first term and third term is 5 times the common difference.

To Find:-

• The first three terms.

Solution:-

Let the first three terms be (a - d) , (a) and and (a + d) respectively.

As, the sum of first three terms is 18

→ (a - d) + (a) + (a + d) = 18

→ a - d + a + a + d = 18

→ a + a + a + d - d = 18

→ 3a = 18

→ a = $\dfrac{18}{3}$

→ a = 6

Given, the product of first term and third term is 5 times the common difference

→ (a - d) (a + d) = 5d

→ a² - d² = 5d

→ (6)² - d² = 5d $\: \: \: \: \: \: \: \: \: \rm [a = 6]$

→ 36 - d² = 5d

→ d² + 5d - 36 = 0

→ d² + 9d - 4d - 36 = 0

→ d(d + 9) - 4(d + 9) = 0

→ (d - 4) + (d + 9) = 0

→ d = 4 and d = -9

If d = 4 and a = 6

The first three terms are 6 - 4 , 6 , 6 + 4

= (2, 6 , 10)

If d = -9 and a = 6

The first three terms are 6 -(-9) , 6 , 6 -9

= 15, 6 , -3

Therefore, the first three terms of the AP are (2, 6, 10) and (15, 6 , -3)