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.
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
Thanks so much Bosnian... It never crossed my mind to introduce a fourth variable. You are a rock star :D
1 answer