Question

8, x, y, 24

The first three terms are geometric.
The last three terms are arithmetic.
Find x and y.

Answer 1

x^2 = 8y

y-x = 24-y
solve to get
8, -8, 8, 24
or
8, 12, 18, 24

Answer 2

relation #1:

x-8 = y-x
y = 2x - 8

relation #2:
y/x = 24/y
y^2 = 24x or x = y^2/24

substitution
y = 2(y^2/24) - 8
12y = y^2 - 96
y^2 - 12y - 96 = 0
solving using the formual I get
x = 7-√33, y = 6 - 3√33
or
x = 7 + √33, y = 6 + 2√33

both case work.
I was expecting some rational answers, are you sure
the question was posted corrrectly?

Answer 3

arggghhhh!

got the whole thing backwards, go with oobleck

(it was still fun doing it the wrong way)

Answer 4

[AI-generated answer — Explain Bot]

To find x and y, we need to analyze the given sequence - 8, x, y, 24 - and apply the information that the first three terms are geometric while the last three terms are arithmetic.

In a geometric sequence, each term is obtained by multiplying the previous term by a constant ratio. Let's assign this ratio as "r" for our sequence.

So, for the geometric part of the sequence:
The second term (x) is obtained by multiplying the first term (8) by the ratio (r): 8 * r = x
The third term (y) is obtained by multiplying the second term (x) by the same ratio (r): x * r = y

In an arithmetic sequence, each term is obtained by adding a constant difference to the previous term. Let's assign this difference as "d" for the arithmetic part of our sequence.

So, for the arithmetic part of the sequence:
The third term (y) is obtained by adding the difference (d) to the second term (x): x + d = y
The fourth term (24) is obtained by adding the same difference (d) to the third term (y): y + d = 24

Now, we have a system of equations:

8 * r = x (Eq. 1)
x * r = y (Eq. 2)
x + d = y (Eq. 3)
y + d = 24 (Eq. 4)

To solve this system, we can use substitution or elimination. Let's solve it using substitution:

From Eq. 3, we have x + d = y.
Rearranging this equation, we get:
x = y - d (Eq. 5)

Substituting Eq. 5 into Eq. 1, we get:
8 * r = y - d (Eq. 6)

Now, we can substitute this value of x into Eq. 2 to solve for y:
(y - d) * r = y

Expanding this equation:
y * r - d * r = y

Rearranging and simplifying:
y - y * r = d * r

Factoring out y:
y(1 - r) = d * r

Dividing both sides by (1 - r):
y = (d * r) / (1 - r) (Eq. 7)

Substituting this value of y into Eq. 4:
(d * r) / (1 - r) + d = 24

Multiplying both sides by (1 - r) to eliminate the denominator:
d * r + d * (1 - r) = 24 * (1 - r)

Expanding and simplifying:
d * r + d - d * r = 24 - 24r

Cancelling the common terms:
d = 24 - 24r

Now, we have expressions for x and d in terms of r:
x = y - d (from Eq. 5)
d = 24 - 24r (from above simplification)

We can substitute the expression for d in terms of r into x = y - d to get x in terms of r:
x = y - (24 - 24r)
x = y + 24r - 24

Finally, we substitute these expressions for x and y into Eq. 2 to get an equation in terms of r:

(y + 24r - 24) * r = y

Expanding and simplifying:
y * r + 24r^2 - 24r = y

Rearranging and canceling the common terms:
24r^2 - 24r = 0

Factoring out 24r:
24r * (r - 1) = 0

Setting each factor to zero:
24r = 0 or r - 1 = 0

From the first factor, we can see that r = 0. However, in a geometric sequence, the common ratio cannot be zero, as it will result in all subsequent terms being zero.

So, let's solve r - 1 = 0:
r = 1

Now that we have the value of r, we can substitute it back into our expressions for x and y to find their values:

From Eq. 7:
y = (d * r) / (1 - r)
y = (24 * 1) / (1 - 1)
y = 24 / 0 (Division by zero is undefined)

As we can see, y is undefined when r=1, which means there is no solution for the given sequence such that the first three terms are geometric and the last three terms are arithmetic.