Question

the sum of three numbers of AP is 27 and their product is 405 find the numbers​

Answer 1

Solution:-

The three consecutive term of AP is ( a - d ) ( a ) ( a + d )

Now ,

The sum of three numbers of AP = 27

=> a - d + a + a + d = 27

=> 3a = 27

=> a = 27/ 3

=> a = 9

Now

The product of three number is 405

( a - d ) × ( a ) × ( a + d ) = 405

Using this identity

(a - b)(a + b ) = ( a² - b² )

We get

( a² - d² )a = 405

We have already find value of a = 9

{(9)² - d²} 9 = 405

(81 - d² ) 9 = 405

729 - 9d² = 405

-9d² = 405 - 729

-9d² = - 324

d² = 36

d = ± 6

Now

Case = 1

a = 9 and d = +6

put the value in this

( a - d ) ( a ) ( a + d )

(9 - 6) , 9 ,( 9 + 6 )

=> 3 , 9 , 15

Case = 2

a = 9 and d = - 6

put the value in this

( a - d ) ( a ) ( a + d )

( 9 - ( - 6), ( 9) , ( 9 - 6 )

=> 15 , 9 , 3

Answer 2

$\sf{\underline{\boxed{\large{\blue{\mathsf{Correct\: Question}}}}}}$

• the sum of three numbers of AP is 27 and their product is 405 find the numbers

_______________________

$\sf{\underline{\boxed{\large{\blue{\mathsf{Solution}}}}}}$

$\sf{\bold{\green{\underline{\underline{Given}}}}}$

• Sum of 3 no. in AP = 27
• Product of 3no. in AP = 405

_______________________

$\sf{\bold{\green{\underline{\underline{To\:Find}}}}}$

• Find the numbers = ??

______________________

$\sf{\bold{\green{\underline{\underline{Solution}}}}}$

• let the no. be x -

let the 3no. be ( x - a ) , x , ( x + a )

• sum of 3no. of AP = 27

$\sf\implies x - a + x + x + a = 27$

$\sf\implies 3x = 27$

$\sf\implies x = \dfrac{27}{3}$

$\sf\implies x = 9$

$\sf{\red{\boxed{\bold{x = 9 }}}}$

• product of 3 no. of AP is 405

$\sf\implies ( x - a ) ( x ) ( x + a ) = 405$

$\sf\implies ( 9 + a )( 9 - a) ( 9) = 405$

$\sf{\red{\boxed{\bold{( a + b ) ( a - b ) = a^2 - b^2 }}}}$

$\sf\implies 81 - a^2 = \dfrac{405}{9}$

$\sf\implies a^2 = 81 - 45$

$\sf\implies a^2 = 36$

$\sf\implies a = 6$

$\sf{\red{\boxed{\bold{a = 6 }}}}$

• finding the no.

First no. = x - a = 9 - 6 = 3

Second no. = x = 9

Third no. = x + a = 9 + 6 = 15

___________________________

$\sf{\bold{\green{\underline{\underline{Answer}}}}}$

• Therefore the no. are 3 , 9 , 15