the following is an AP 9,X,Y,Z,25

1.find the values of x,y,z.

There's a difference of 4 between each term.

9, 13, . . .

To find the values of x, y, and z in the arithmetic progression (AP) 9, x, y, z, 25, we need to use the properties of an arithmetic progression.

In an arithmetic progression, the difference between any two consecutive terms is constant.

Let's calculate the common difference, d:

d = (second term - first term) = (x - 9)
Since the common difference is constant, the difference between the second and third terms will also be (x - 9), and the difference between the third and fourth terms will also be (x - 9).

Now, we can set up two equations to find the values of x, y, and z:

Equation 1: y = x + (x - 9)
Equation 2: z = y + (x - 9)

Since the fourth term of the AP is 25, we can also set up the equation:

Equation 3: 25 = z + (x - 9)

Combining equations 1, 2, and 3, we can solve for x, y, and z.

First, substitute Equation 2 into Equation 3:
25 = (y + x - 9) + (x - 9)
25 = y + 2x - 18

Next, substitute Equation 1 into the above equation:
25 = (x + (x - 9)) + 2x - 18

Now, simplify and solve for x:
25 = 2x + (2x - 18)
25 = 4x - 18
4x = 43
x = 10.75

Now that we have found the value of x, we can substitute it back into Equation 1 to find y:
y = x + (x - 9)
y = 10.75 + (10.75 - 9)
y = 10.75 + 1.75
y = 12.5

Similarly, substituting the values of x and y into Equation 2, we can find z:
z = y + (x - 9)
z = 12.5 + (10.75 - 9)
z = 12.5 + 1.75
z = 14.25

Therefore, the values of x, y, and z in the AP 9, x, y, z, 25 are:

x = 10.75
y = 12.5
z = 14.25