what is x if the numbers a+x, b+x, c+x are terms of geometric progression ?

Since the given terms are the consecutive terms of a
geometric progression, we'll write the relation between
them:

(b + x)^2 =
(a+x)(c+x)

We'll expand the square from the left side and
we'll remove the brackets from the right side:

b^2 + 2bx +
x^2 = ac + ax + cx + x^2

We'll eliminate
x^2:

b^2 + 2bx = ac +
x(a+c)

We'll move the terms in x to the left side and the
terms without x, to the right side:

2bx - x(a+c) = ac -
b^2

We'll factorize by x and we'll
get:

x(2b - a - c) = ac -
b^2

We'll divide by (2b - a -
c):

x = (b^2 - ac)/(a + c -
2b)