Fourth Proportional to x, 2x, and (x + 1)

To find the fourth proportional to x, 2x, and (x + 1), we need to understand what a fourth proportional is. In a proportion, four quantities are related in such a way that the fourth quantity is directly proportional to the other three. This means that if we have three numbers, we can find a fourth number that is proportional to them.

Let's break down the given numbers and find the fourth proportional.

Given numbers:
- x
- 2x
- (x + 1)

Step 1: Write the proportion equation:
We can write the proportion equation as follows:
x : 2x = (x + 1) : y

Here, y represents the fourth proportional number that we need to find.

Step 2: Solve the proportion equation:
To solve the proportion equation, we can cross-multiply and then solve for y.

x * y = 2x * (x + 1)

Simplifying the equation, we get:
xy = 2x^2 + 2x

Step 3: Solve for y:
To find the value of y, we need to isolate it on one side of the equation. Let's rearrange the equation:

xy - 2x^2 - 2x = 0

Factor out x from the left side of the equation:
x(y - 2x - 2) = 0

Now, we have two possible solutions:
1. x = 0
2. y - 2x - 2 = 0

Case 1: x = 0
If x = 0, then the equation becomes:
0(y - 2(0) - 2) = 0
0(y - 2) = 0

This means that y can take any value as long as y - 2 = 0, which gives us y = 2.

Case 2: y - 2x - 2 = 0
If y - 2x - 2 = 0, then we can solve for y:
y = 2x + 2

In this case, y is dependent on the value of x. As x changes, y will also change accordingly.

Conclusion:
The fourth proportional to x, 2x, and (x + 1) can have two possible solutions:
1. If x = 0, then the fourth proportional is y = 2.
2. If y - 2x - 2 = 0, then the fourth proportional is y = 2x + 2, where y is dependent on the value of x.