6,x,y,16 is a sequence of numbers.6,x.y form an arithmetic sequence.x,y,16 form a geometric sequence. Detrmine the values of x and y

for the AS

x-6 = y-x
2x-y = 6 or y = 2x-6

for the GS
y/x = 16/y
y^2 = 16x
sub in the 1st
(2x-6)^2 = 16x
4x^2 - 24x + 36 = 16x
4x^2 -40x + 36 = 0
x^2 - 10x + 9 = 0
(x-1)(x-9) = 0
x = 1 or x = 9

if x = 1, y = -4
if x = 9 , y = 12

check:
1st case: 6, 1, -4, 16 , the first 3 are AS, the last 3 are GS
2nd case: 6, 9 , 12, 16 , again, the given conditions are met

THEY DIDN'T MULTIPLY BY 2 THATS (2x-6)² FOR y²

To determine the values of x and y in the given sequence, we will use the information that the numbers 6, x, and y form an arithmetic sequence, and x, y, and 16 form a geometric sequence.

First, let's find the common difference (d) of the arithmetic sequence.
The common difference (d) is the difference between any two consecutive terms in an arithmetic sequence.

Using the first two terms of the arithmetic sequence, we have:
x - 6 = d

Now, let's find the common ratio (r) of the geometric sequence.
The common ratio (r) is the ratio between any two consecutive terms in a geometric sequence.

Using the last two terms of the geometric sequence, we have:
16/y = r

From the given information, we know that x, y, and 16 form a geometric sequence. Hence, we can express y in terms of x:
y = xr

Substituting the above expression for y into the equation 16/y = r, we have:
16/(xr) = r

To simplify this equation, multiply both sides by xr:
16 = r^2 * x

Now, we have two equations:
1) x - 6 = d
2) 16 = r^2 * x

To solve these equations simultaneously, let's substitute x - 6 from the first equation into the second equation:
16 = r^2 * (x - 6)

Expanding this equation:
16 = r^2 * x - 6r^2

Rearrange the equation:
6r^2 - r^2 * x = -16

Now, we have an equation in terms of x and r. Let's rearrange it to solve for x:
x * r^2 - 6r^2 = -16

Factor out r^2:
r^2 * (x - 6) = -16

Divide both sides by (x - 6):
r^2 = -16 / (x - 6)

We know that the common ratio (r) must be a non-zero real number. Hence, -16/(x - 6) cannot be zero.
Therefore, (x - 6) must not equal zero:
x - 6 ≠ 0

Solving the inequality:
x ≠ 6

So, the value of x cannot be 6.

To summarize:
We have determined that the value of x cannot be 6 due to the constraint that the common ratio (r) cannot be zero. However, without additional information or constraints, it is not possible to determine the values of x and y uniquely.

why did multiply by 2 when you got 2x-6