What would be the value of x so that x+2, 3x-2, 7x-12 will form an arithmetic sequence

To solve this problem, we need to find the value of \( x \) such that the terms \( x + 2 \), \( 3x - 2 \), and \( 7x - 12 \) form an arithmetic sequence. 

In an arithmetic sequence, the difference between consecutive terms is constant. This means:

\[
(3x - 2) - (x + 2) = (7x - 12) - (3x - 2)
\]

Let's simplify both sides of the equation:

1. Calculate the left side:

\[
(3x - 2) - (x + 2) = 3x - 2 - x - 2 = 2x - 4
\]

2. Calculate the right side:

\[
(7x - 12) - (3x - 2) = 7x - 12 - 3x + 2 = 4x - 10
\]

Now, we set these two expressions equal to each other:

\[
2x - 4 = 4x - 10
\]

Solve for \( x \) by first isolating \( x \) on one side:

\[
2x - 4 = 4x - 10 \implies -4 + 10 = 4x - 2x \implies 6 = 2x \implies x = 3
\]

Thus, the value of \( x \) is \( \boxed{3} \).

### Verification
Substitute \( x = 3 \) back into the expressions for the sequence terms:

1. \( x + 2 = 3 + 2 = 5 \)
2. \( 3x - 2 = 3(3) - 2 = 9 - 2 = 7 \)
3. \( 7x - 12 = 7(3) - 12 = 21 - 12 = 9 \)

The terms are \( 5, 7, 9 \), which form an arithmetic sequence with a common difference of 2. Thus, the solution is correct.

Would you like to explore more about arithmetic sequences, or have any questions?

### Related Questions
1. What is an arithmetic sequence and how can it be identified?
2. How do you find the common difference in an arithmetic sequence?
3. What is the formula for the \( n \)-th term of an arithmetic sequence?
4. How can we find the sum of the first \( n \) terms in an arithmetic sequence?
5. How do we determine if three given numbers form a geometric sequence?

#### Tip
Remember, for any arithmetic sequence, the middle term is always the average of the first and the last terms.

What would be the value of x so that x + 2, 3x - 2, 7x - 12 will form an arithmetic sequence?

Discover how to find the value of x that makes x + 2, 3x - 2, and 7x - 12 form an arithmetic sequence. Learn the concepts of arithmetic sequences and how to solve linear equations, perfect for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation