if x,y,z represents consecutive terms in an Arithmetic progression, prove that y is = (x+z)/2

State that the difference to be 'b'

y = x + b
b = y - x. ..(1)

z = y + b = x + 2b
b = (z - x)/2. ..(2)

Combining equations (1) and (2):
b = y - x = (z - x)/2
y = (z + x)/2

To prove that y = (x + z)/2 for consecutive terms x, y, and z in an arithmetic progression, we need to demonstrate that the common difference between any two consecutive terms is equal. In other words, we have to show that z - y = y - x.

Let's start by considering the difference between the second term (y) and the first term (x) in the arithmetic progression:
y - x = (x + d) - x
= d

Here, d represents the common difference between any two consecutive terms.

Next, let's examine the difference between the third term (z) and the second term (y):
z - y = (y + d) - y
= d

We can see that the difference between the second and third terms is also equal to d.

Since both z - y and y - x are equal to d, we can conclude that z - y = y - x. This implies that y is the average of x and z:

y = (x + z)/2

Hence, we have proven that if x, y, and z are consecutive terms in an arithmetic progression, then y = (x + z)/2.