Question

Find the three number of AP whose Sum &
product are 15 and105 respectively
​

Answer 1

Let the required terms of that A.P. be a - d, a and a + d.

According to the question : -

• Sum of three numbers = 15
• Product of three numbers = 105

On substituting the algebraic values instead of sentence : -

= > Sum of terms = 15

= > ( a - d ) + a + ( a + d ) = 15

= > a - d + a + a + d = 15

= > 3a = 15

= > a = 15 / 3

= > a = 5

Now, also, given

= > Their product = 105

= > ( a - d )( a )( a + d ) = 105

= > ( 5 - d )5( 5 + d ) = 105 { from above, a = 5 }

= > ( 5 - d )( 5 + d ) = 105 / 5

= > ( 5 )^2 - d^2 = 21 { ( a - b )( a + b ) = a^2 - b^2 }

= > 25 - d^2 = 21

= > 25 - 21 = d^2

= > 4 = d^2

= > ( ± 2 )^2 = d^2

= > ± 2 = d

Case 1 : Where numeric value of d is 2.

Terms are :

• a - d = 5 - 2 = 3

• 5

• a + d = 5 + 2 = 7

Case 2 : Where numeric value of d is - 2.

Terms are :

• a - d = 5 - ( - 2 ) = 5 + 2 = 7

• 5

• a + d = 5 + ( - 2 ) = 5 - 2 = 3

Hence the required terms of that AP are 3 , 5 and 7 or 7 , 5 and 3.