if 8; 2x; 2y form an arithmetic sequence and 2x; 2y; 36 form a geometric sequence determine the values of x and y

I feel like I am going around in circles on this problem.

For the AP i have the following formulas:
(1) d=(2y-8)/2
(2) d=2x-8
(3.1) (2y-8)/2=2x-8
or
(3.2) (2y-8)/2-(2x-8)=8

For the GP:
(1) 4y^2=72x

I know i am so how meant to use simultaneous equations to solve this , but I just cant seem to see how?

Any help with this problem would be appreciated.

Many thanks

2 answers

For the AP:

2 x = 8 + d

2 y = 2 x + d = 8 + d + d = 8 + 2 d

2 y = 8 + 2 d

For the GP:

2 y = 36 / q

2 x = 2 y / q = ( 36 / q ) / q = 36 / q ^ 2

So:

2 x = 2 x

8 + d = 36 / q ^ 2

2 y = 2 y

8 + 2 d = 36 / q

Now you must solve system:

8 + d = 36 / q ^ 2

8 + 2 d = 36 / q

8 + 2 d = 36 / q Subtract 8 to both sides

8 + 2 d - 8 = 36 / q - 8

2 d = 36 / q - 8 Divide both sides by 2

d = ( 36 / q ) / 2 - 8 / 2

d = ( 18 / q ) - 4

Replace this value in equation:

8 + d = 36 / q ^ 2

8 + 18 / q - 4 = 36 / q ^ 2

4 + 18 / q = 36 / q ^ 2 Multiply bothsides by q ^ 2

4 q ^ 2 + 18 q ^ 2 / q = 36 q ^ 2 / q ^ 2

4 q ^ 2 + 18 q = 36 Subtract 36 to both sides

4 q ^ 2 + 18 q - 36 = 36 - 36

4 q ^ 2 + 18 q - 36 = 0

Try to solve this equation.

The solutions are :

q = - 6 and q = 3 / 2

For q = - 6:

d = 18 / q - 4

d = 18 / - 6 - 4

d = - 3 - 4

d = - 7

For q = 3 / 2:

d = 18 / ( 3 / 2 ) - 4

d = 18 * 2 / 3 - 4

d = 36 / 3 - 4

d = 12 - 4

d = 8

For q = - 6 and d = - 7

2 x = 8 + d

2 x = 8 + ( - 7 )

2 x = 8 - 7

2 x = 1 Divide both sides by 2

x = 1 / 2


2 y = 8 + 2 d

2 y = 8 + 2 * ( - 7 )

2 y = 8 - 14

2 y = - 6 Divide both sides by 2

y = - 6 / 2

y = - 3


For q = 3 / 2 and d = 8

2 x = 8 + d

2 x = 8 + 8

2 x = 16 Divide both sides by 2

x = 16 / 2

x = 8

2 y = 8 + 2 d

2 y = 8 + 2 * 8

2 y = 8 + 16

2 y = 24 Divide both sides by 2

y = 24 / 2

y = 12


All this mean you have 2 set of solutions:

1)

q = - 6 , d = - 7 , x = 1 / 2 , y = - 3

2)

q = 3 / 2 , d = 8 , x = 8 , y = 12


1 solution:

AP:

8 , 8 + d , 8 + 2 d

8 , 8 + ( - 6 ) , 8 + 2 * ( - 6 )

8 , 8 - 6 , 8 - 12

8 , 2 , - 4

GP:

2 x , 2 y , 2 y * q

2 * 1 / 2 , 2 * ( - 3 ) , 2 * ( - 3 ) * ( - 6 )

1 , - 6 , 2 * 18

1 , - 6 , 36

Or GP:

1 , 1 * q , 1 * q ^ 2

1 , 1 * ( - 6 ) , 1 * ( - 6 ) ^ 2

1 , - 6 , 1 * 36

1 , - 6 , 36

2 solution:

q = 3 / 2 , d = 8 , x = 8 , y = 12

AP:

8 , 8 + d , 8 + 2 d

8 , 8 + 8 , 8 + 2 * 8

8 , 16 , 8 + 16

8 , 16 , 24

GP:

2 x , 2 y , 2 y * q

2 * 8 , 2 * 12 , 2 * 12 * 3 / 2

16 , 24 , 24 * 3 / 2

16 , 24 , 72 / 2

16 , 24 , 36

Or GP:

16 , 16 * 3 / 2 , 16 * ( 3 / 2 ) ^ 2

16 , 48 / 2 , 16 * 9 / 4

16 , 24 , 144 / 4

16 , 24 , 36

I hope I was not too verbose.
Thanks so much Bosnian... It never crossed my mind to introduce a fourth variable. You are a rock star :D