The first three terms of the sequence 6;x;y;27 form an arithmetic progression and the last three terms form a geometric progression. Determine the values of x and y
x = 15 and y = 18
x = 6 + a
y = 6 + 2 a
y = x r
27 = x r^2
x r = 6 + 2 a
so
(6+a) r = 6 + 2a so r = (6+2a) / (6+a)
and
27 = (6+a) r^2
27 = (6+a) (6+2a)^2 / (6+a)^2 = (6^2 + 24 a + 4a^2)/(6+a)
162 + 27 a = 4 a^2 + 24 a + 36
4 a^2 -3 a -126 = 0
positive solution of quadratic for a = 6
so
x = 12
y = 18
or
y - x = x - 6
y = 2x - 6
y^2 = 27x
(2x-6)^2 = 27x
4x^2 - 24x + 36 = 27x
4x^2 - 51x + 36 = 0
(x - 12)(4x - 3) = 0
x = 12 or x = 3/4
if x =12, y = 18
if x = 3/4 , y = -9/2
To determine the values of x and y in the given sequence, we need to use the information that the first three terms form an arithmetic progression and the last three terms form a geometric progression.
Let's start by considering the fact that the first three terms form an arithmetic progression. In an arithmetic progression, the difference between consecutive terms is constant.
Let's assume the common difference in the arithmetic progression is 'd'. So, the second term would be 6 + d, and the third term would be (6 + d) + d = 6 + 2d.
Now, let's move on to the fact that the last three terms form a geometric progression. In a geometric progression, the ratio between consecutive terms is constant.
Let's assume that the common ratio in the geometric progression is 'r'. So, the third term would be y * r, and the second-to-last term would be y * r^2.
Now, we can use the information we have to set up two equations and solve for x and y.
Equation 1: (6 + d) - 6 = (6 + 2d) - (6 + d)
Simplifying Equation 1: d = 2d - d
Resulting in: d = 0
Since d = 0, the terms in the arithmetic progression are actually equal. Therefore, the first three terms of the sequence are all equal to 6.
Now let's move on to find the common ratio in the geometric progression.
Equation 2: (27 / y) = y * r^2 / (y * r)
Simplifying Equation 2: 27 / y = r
We know that the common ratio is equal to 27 / y. Therefore, we have:
27 / y = r
Since the common ratio is constant, we can set up another equation using the second and third terms:
y * r = 6 + 2d
Substituting r with 27 / y:
y * (27 / y) = 6 + 2 * 0
27 = 6
The equation simplifies to 27 = 6, which is not true.
Hence, there is no solution that satisfies both conditions.