I would need help with example: The sum of three consecutive terms of geometric progression is 9. The first number with no change, the second number plus 12 and the third number minus 3, are the 3 consequtive terms of arithmetic progression. What are the values of original 3 terms?

I can not move with it. I would be grateful for any help.
Thank you.

Let the 3 terms as a GP be

a , ar, and ar^2
a + ar + ar^2 = 9
a(r^2 + r + 1) = 9 , #1

new 3 terms:
a
ar+12
ar^2 - 3

now they are in a AS
ar+12 - a = ar^2-3 - (ar+12)
ar - a + 12 = ar^2 - ar - 15
ar^2 - 2ar + a = 27
a(r^2 - 2r + 1) = 27 , #2

divide #2 by #1

(r^2 - 2r + 1)/(r^2 + r + 1 = 27/9 = 3
3r^2 + 3r + 3 = r^2 - 2r + 1
2r^2 + 5r + 2 = 0
(2r + 1)(r + 2) = 0

r = -1/2 or r = -2

if r = -2, in #1,
a(4-2+1) = 9
a = 9/3=3
The 3 original terms are 3 , -6 , 12

if r = -1/2 , in #1
a(1/4 - 1/2 + 1) = 9
a((3/4) = 9
a = 12

or the 3 original terms were 12 , -6 , 3

I did get first two equations but didn´t know how to continue. Thank you very much for help.

To solve this problem, let's break it down step by step.

Step 1: Understand the problem
We are given a situation where there are three consecutive terms in a geometric progression whose sum is 9. Additionally, the second term plus 12 and the third term minus 3 form an arithmetic progression. We need to find the values of the original three terms.

Step 2: Set up an equation for the geometric progression
Let's assume the first term of the geometric progression is 'a' and the common ratio is 'r'. Since the three terms are consecutive, the second term would be 'ar' and the third term would be 'ar^2'.

The sum of the three consecutive terms is given by the formula:
sum = (a + ar + ar^2) = 9

Step 3: Set up equations for the arithmetic progression
The second term plus 12 forms an arithmetic progression, so we can write the equation:
(ar) + 12 = (ar^2) - 3

Simplifying this equation gives us:
12 = (ar^2) - (ar) - 3
(ar^2) - (ar) - 15 = 0

Step 4: Solve the quadratic equation
To solve the quadratic equation (ar^2) - (ar) - 15 = 0, we can use either factoring or the quadratic formula. Let's use the quadratic formula:

r = ( -b ± √(b^2 - 4ac) ) / (2a)

In our case, a = 1, b = -1, and c = -15.

Substituting these values into the quadratic formula, we get:
r = ( -(-1) ± √((-1)^2 - 4(1)(-15)) ) / (2(1))
r = (1 ± √(1 + 60)) / 2
r = (1 ± √61) / 2

Step 5: Find the values of 'a', 'r', and the original three terms
Now that we have found the possible values of 'r', we can substitute them back into the equation for the geometric progression to find the corresponding values of 'a' and the original three terms.

Let's consider the positive value of 'r':
r = (1 + √61) / 2

Substituting this value into the equation for the geometric progression (sum = 9), we get:
(a + a(1 + √61) / 2 + a(1 + √61) / 2)^2 = 9

Simplifying this equation will give us the positive value of 'a' and the corresponding values of the original three terms.

However, without further information, it is not possible to determine the exact values of the original three terms. Additional information or constraints may be required to solve the problem completely.