Find the value of x for which the numbers x+1, x+3, x+7, are in geometric progression.

If in a GP, then

(x+3)/(x+1) = (x+7)/(x+3)
x^2 + 6x + 9 = x^2 + 8x + 7
-2x = -2
x=1

check:
numbers would be 2 , 4, 8 which are in GP with a common ratio of 2

Why did the number cross the road? Because it heard there was a geometric progression party happening on the other side!

To find the value of x for which the numbers x+1, x+3, and x+7 are in geometric progression, we need to look at their ratios. In a geometric progression, the ratio between any two consecutive terms is always the same. Let's write the ratios and set them equal to each other:

(x+3)/(x+1) = (x+7)/(x+3)

Now let's cross-multiply and simplify:

(x+3)^2 = (x+1)(x+7)
x^2 + 6x + 9 = x^2 + 8x + 7

Now let's solve for x:

6x - 8x = 7 - 9
-2x = -2
x = 1

So, the value of x for which the numbers x+1, x+3, and x+7 are in geometric progression is x = 1.

To find the value of x for which the numbers x+1, x+3, x+7 are in a geometric progression, we can use the property of a geometric progression where each term is obtained by multiplying the previous term by a fixed ratio.

Let's assume the common ratio is denoted by 'r'.

According to the definition of a geometric progression:
(x+3) / (x+1) = (x+7) / (x+3)

We can cross-multiply to get rid of the fractions:
(x+3) * (x+3) = (x+1) * (x+7)

Expanding both sides of the equation:
x^2 + 6x + 9 = x^2 + 8x + 7

Simplifying the equation:
x^2 - x^2 + 6x - 8x = 7 - 9
-2x = -2

Dividing both sides of the equation by -2:
x = 1

Therefore, the value of x for which the numbers x+1, x+3, x+7 are in geometric progression is x = 1.

To find the value of x for which the numbers x+1, x+3, x+7 are in geometric progression, we can use the definition of a geometric progression.

In a geometric progression, each term after the first is obtained by multiplying the previous term by a constant called the common ratio.

Let's denote the common ratio as r. Using this information, we have the following:

(x+3) / (x+1) = (x+7) / (x+3)

To solve this equation, we can begin by cross-multiplying:

(x+3)(x+3) = (x+1)(x+7)

Expanding both sides of the equation:

x^2 + 6x + 9 = x^2 + 8x + 7

Simplifying by canceling out the common terms:

6x - 8x = 7 - 9

-2x = -2

Dividing both sides by -2 to solve for x:

x = 1

Therefore, the value of x for which the numbers x+1, x+3, x+7 are in geometric progression is x = 1.