The first three terms of the sequence -8,x,y,72... form an arithmetic sequence, while the second, third and fourth terms form a geometric sequence. Determine x and y.
y-x = x+8
y/x = 72/y
y = 2x+8
(2x+8)/x = 72/(2x+8)
4x^2+32x+64 = 72x
4x^2-40x+64 = 0
x^2 - 10x + 16 = 0
(x-2)(x-8) = 0
x=2,8
y=12,24
so, we have
-8,2,12,72
or
-8,8,24,72
thanks:-)
To determine the values of x and y, we need to first find the common difference (d) of the arithmetic sequence and the common ratio (r) of the geometric sequence.
For the arithmetic sequence, we know that the difference between any two consecutive terms is constant. So, we can subtract the first term from the second term and the second term from the third term to find the common difference.
Second term - First term = x - (-8) = x + 8
Third term - Second term = y - x
Since these differences should be equal, we have the equation:
x + 8 = y - x
For the geometric sequence, we know that the ratio between any two consecutive terms is constant. So, we can divide the third term by the second term and the second term by the first term to find the common ratio.
Third term / Second term = 72 / y = r
Second term / First term = y / x = r
Since these ratios should be equal, we have the equation:
72 / y = y / x
Now we have two equations:
1) x + 8 = y - x
2) 72 / y = y / x
To solve these equations, we can use substitution. From equation 1), we can rearrange it to get x in terms of y:
2x = y - 8
x = (y - 8) / 2
Substitute this value of x into equation 2):
72 / y = y / ((y - 8) / 2)
Now we can solve for y by cross-multiplying:
72 * 2 = y * (y - 8)
144 = y^2 - 8y
0 = y^2 - 8y - 144
Now we can factor this quadratic equation:
0 = (y - 18)(y + 8)
Setting each factor to zero, we have two possible values for y:
y - 18 = 0 --> y = 18
y + 8 = 0 --> y = -8
If y = 18, substitute it back into equation 1) to find the value of x:
x = (18 - 8) / 2
x = 10 / 2
x = 5
So, when y = 18, the values of x and y that satisfy both conditions are x = 5 and y = 18.
If y = -8, substitute it back into equation 1) to find the value of x:
x = (-8 - 8) / 2
x = -16 / 2
x = -8
So, when y = -8, the values of x and y that satisfy both conditions are x = -8 and y = -8.
Therefore, the two possible solutions for x and y are (5, 18) and (-8, -8).