find X if the numbers X+3, 5x-3, and 7x+3 are three consecutive terms of a G.P of positive terms. with this value X and given the X+3, 5x -3 and 7x +3 are the third, fourth and fifth terms of the G.P.. find the sum of the first 8 terms of the progression.

If GP, then

(5x-3)/(x+3) = (7x+3)/(5x-3)
25x^2 - 30x + 9 = 7x^2 + 24x + 9
18x^2 - 54x = 0
6x(3x - 9) = 0
x= 0 or x = 3

if x = 0
3rd term = 3
4th term = -3
5th term = 3
..
1st term = 3
2nd term = -3
sum of any even number of terms = 0

if x = 3,
3rd term = 6
4th term = 12
5th term = 24
so r = 2
t(3) = ar^2 = 6
a = 3/2

sum(8) = a(r^8 - 1)/(r-1)
= (3/2)((2^8 - 1)/(3/2 - 1)
= (3/2)(255)/(1/2)
= 765

Well, well, well. Looks like we've got ourselves a geometric progression conundrum. Let's get down to business, shall we?

In a geometric progression (G.P), each term is found by multiplying the previous term by a common ratio. So, to find the common ratio, we can set up an equation using the given terms:

(X+3) * r = (5x-3) ----(1)
(5x-3) * r = (7x+3) ----(2)

Since (X+3) is the third term, and (5x-3) is the fourth term, we'll focus on these two equations.

Let's solve equation (1) for the value of r. Expanding and simplifying, we get:

Xr + 3r = 5x - 3

Now, let's solve equation (2) by substituting the value of (5x-3) from equation (1):

(5x - 3) * r = 7x + 3
(5x - 3)(Xr + 3r) = 7x + 3

Expanding and simplifying again, we get:

5Xr + 15r - 3Xr - 9r = 7x + 3

Combine like terms:

(5Xr - 3Xr) + (15r - 9r) = 7x + 3
2Xr + 6r = 7x + 3

Now, substituting the value of (Xr + 3r) from equation (1):

2(X+3) + 6r = 7x + 3
2X + 6 + 6r = 7x + 3

Rearranging and simplifying:

2X - 7x = -3 - 6 - 6r + 3
2X - 7x = -6 - 6r

We've got one wild equation there, but don't worry, Clow Bot is here to save the day! Now we can solve it to find our value of X.

Unfortunately, my clown calculations indicate that we cannot determine the exact value of X based on the given information. The equation does not simplify further, so we'll have to keep the solution in terms of X and the common ratio.

As for finding the sum of the first 8 terms of the progression, without the exact value of X and the common ratio, my comical abilities cannot help in this case. It seems I've hit a brick wall. I apologize for not being able to provide a precise answer.

Well, you got mistakes in the second part

To find the common ratio (r) of a geometric progression (G.P), we can use the formula:

r = (n-th term) / (previous term)

Let's find the common ratio (r) of the given G.P using the third and fourth terms:
r = (5x - 3) / (x + 3)

Since the third, fourth, and fifth terms of the G.P are given as (x + 3), (5x - 3), and (7x + 3) respectively, we can also write the common ratio as:
r = (7x + 3) / (5x - 3)

Now that we have two expressions for the common ratio, we can equate them and solve for x:

(x + 3) / (5x - 3) = (7x + 3) / (x + 3)

Cross-multiplying:

(x + 3)(x + 3) = (7x + 3)(5x - 3)

Expanding and simplifying:

x^2 + 6x + 9 = 35x^2 - 6

Rearranging:

34x^2 - x^2 + 6x + 6 - 9 = 0
33x^2 + 6x - 3 = 0

This quadratic equation can be solved using factoring, completing the square, or using the quadratic formula. For simplicity, let's use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

Plugging in the values:

x = (-6 ± √((6^2) - 4(33)(-3))) / (2(33))

x = (-6 ± √(36 + 396)) / 66

x = (-6 ± √432) / 66

x = (-6 ± √(36 * 12)) / 66

x = (-6 ± 6√3) / 66

Simplifying further:

x = (-1 ± √3) / 11

So, there are two possible values for x: (-1 + √3) / 11 and (-1 - √3) / 11.

Now, let's find the sum of the first 8 terms of the given G.P.

To find the sum of the first n terms of a geometric progression, you can use the formula:

Sn = a * (1 - r^n) / (1 - r)

Where Sn is the sum of the first n terms, a is the first term, and r is the common ratio.

Given that the third term (a3) is (x + 3), we have:

a = (x + 3)

And the common ratio (r) is:

r = (5x - 3) / (x + 3) (which is the same as (7x + 3) / (5x - 3))

Substituting these values, we have:

S8 = (x + 3) * (1 - [(5x - 3) / (x + 3)]^8) / (1 - (5x - 3) / (x + 3))

Simplifying further:

S8 = (x + 3) * (1 - [(5x - 3)^8 / (x + 3)^8]) / (1 - (5x - 3) / (x + 3))

Since we have the two possible values of x, you can substitute each value separately into the formula to find the sum of the first 8 terms for each case.

The first answer is ok, but for the second answer I got 384 and the book says it's 382.5. I'm really confused