Math, asked by hemantraj888666, 11 months ago

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

Answers

Answered by Anonymous
2

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

Answered by InfiniteSoul
6

\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

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