Question

Three positive integers x , y , z , are in AP such that sum of x, y and z is 33 and product of x, y and z is 1155 then the value of x is

Answer 1

In the above Question, the following information is given -

Three positive integers x , y , z , are in AP such that sum of x, y and z is 33 and product of x, y and z is 1155 .

To find -

The value of x is -

Solution -

Here ,

Three positive integers x , y , z , are in AP such that sum of x, y and z is 33 and product of x, y and z is 1155 .

Let y = y

x = ( y - d )

x = ( y + d )

Now ,

Three positive integers x , y , z , are in AP such that sum of x, y and z is 33

So ,

( y - d ) + y + ( y + d ) = 33

=> 3y = 33

=> y = 11

Product -

=> ( y - d )( y + d ) y

=> ( 11 - d )( 11 + d ) × 11

=> ( 121 - d² ) × 11

=> 1331 - 11d²

But , this is equal to 1155 .

So ,

1331 - 11d² = 1155

=> 11 d² = 176

=> d² = 16

=> d = $\neq$ 4

Thus , x can be 15 as well as 7 .

This is the required answer .

__________________________

Answer 2

◘ Given ◘

Three positive integers x, y, z are in AP such that sum of x, y and z is 33 and product of x, y and z is 1155.

• x + y + z = 33     . . . (i)

• xyz = 1155         . . . (ii)

◘ To Find ◘

The value of x.

◘ Solution ◘

Let the common difference of the AP be d.

We know,

$\sf{\color{red}{d=y-x=z-y}}$

Calculating further :-

$\sf{\implies y+y=z+x}\\\\\sf{\implies y=\dfrac{z+x}{2} }$

Putting the value in (i) :-

$\sf{\implies x+\dfrac{x+z}{2}+z=33 }\\\\\sf{\implies \dfrac{2x+x+z+2z}{2}=33 }\\\\\sf{\implies 3x+3z=66}\\\\\sf{\implies x+z=22}$

We know,

$\sf{ y=\dfrac{z+x}{2} }\\\\\sf{\implies 2y=z+x }$

From above, we get :-

2y = 22

⇒y = 11

$\rule{150}2$

• z = 22 - x
• y = 11

xyz = 1155

$\sf{\implies x(22-x)11=1155}\\\\\sf{\implies 22x-x^2=\dfrac{1155}{11}}\\\\\sf{\implies -x^2+22x=105}\\\\\sf{\implies x^2-22x+105=0}\\\\\sf{\implies x^2-15x-7x+105=0}\\\\\sf{\implies x(x-15)-7(x-15)=0}\\\\\sf{\implies (x-15)(x-7)=0}\\\\\sf{\implies x=15\:\:or\:\:x=7}$

Therefore, the value of x is either 15 or 7.

$\underline{\rule{180}2}$

◘ Additional Info ◘

For the different values of x,

→ The value of z = 7 or 15.

So, the AP is

• 15, 11, 7
• 7, 11, 15