Question

the sum of three numbers in AP is 18.if the product of first and third number is five times the common difference, find the number​

Answer 1

SOLUTION:-

Let the no. (a - d), a, (a + d), be in A.P.

common difference = d

Condition1:-

The sum three numbers in AP is 18.

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

=> a - d + a + a + d = 18

=> 3a = 18

=> a = 18/3

=> a = 6

Condition2:-

The product of first and third number is five times the common difference.

=> (a - d) × (a + d) = 5 × d

=> a(a + d) - d(a + d) = 5d

=> a² + ad - ad - d² = 5d

=> a² - d² = 5d

=> (6)² - d² = 5d

=> 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) = 0 or (d + 9) = 0

=> d - 4 = 0 or d + 9 = 0

=> d = 4 or d = -9

Here the value for d is 4 or - 9

• The number =

(a - d) = 6 - 4 = 2

a = 6

(a + d) = 6 + 4 = 10

so, number = 2, 6, 10

or

(a - d) = 6 - 9 = -3

a = 6

(a + d) = 6 + 9 = 15

so, number = -3, 6, 15