Find x so that x+1, x-1, and 2x are terms of an arithmetic sequence. Write out the terms

For \(x+1\), \(x-1\), and \(2x\) to be terms of an arithmetic sequence, the difference between consecutive terms must be the same. This gives us the condition:

\[
(x - 1) - (x + 1) = 2x - (x - 1)
\]

### Step 1: Simplify the equation
The first term on the left side simplifies as:
\[
(x - 1) - (x + 1) = x - 1 - x - 1 = -2
\]

The right side simplifies as:
\[
2x - (x - 1) = 2x - x + 1 = x + 1
\]

### Step 2: Set up the equation
Now we equate the two expressions:
\[
-2 = x + 1
\]

### Step 3: Solve for \(x\)
Subtract 1 from both sides:
\[
-2 - 1 = x
\]
\[
x = -3
\]

### Step 4: Verify the terms
Substitute \(x = -3\) into the terms:
- \(x + 1 = -3 + 1 = -2\)
- \(x - 1 = -3 - 1 = -4\)
- \(2x = 2(-3) = -6\)

The terms are \(-2\), \(-4\), and \(-6\). The common difference is \(-2\), confirming that the terms form an arithmetic sequence.

Thus, \(x = -3\), and the terms are \(-2\), \(-4\), and \(-6\).

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### Related Questions:
1. What are the general conditions for numbers to form an arithmetic sequence?
2. How would the problem change if the sequence was geometric?
3. How do you find the sum of an arithmetic sequence?
4. Can an arithmetic sequence have increasing terms if one term is negative?
5. How does solving for terms in an arithmetic sequence apply to real-world scenarios?

### Tip:
When working with sequences, always check for the common difference (arithmetic) or ratio (geometric) to confirm the type of sequence.

Find x so that x+1, x-1, and 2x are terms of an arithmetic sequence. Write out the terms.

Learn how to solve for x so that x+1, x-1, and 2x form an arithmetic sequence. The solution explores algebraic methods and the properties of arithmetic sequences. Suitable for students in Grades 8-10.

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