Solve for the sum of the first five terms of the arithmetic sequence 2x, 3x - 2, 4x - 4

To solve for the sum of the first five terms of the given arithmetic sequence, we first need to verify the common difference and then use the formula for the sum of an arithmetic sequence.

### Step 1: Verify the common difference
Given the arithmetic sequence:
- First term (\(a_1\)) = \(2x\)
- Second term (\(a_2\)) = \(3x - 2\)
- Third term (\(a_3\)) = \(4x - 4\)

To find the common difference (\(d\)), subtract the first term from the second term, and the second term from the third term:

\[
d = a_2 - a_1 = (3x - 2) - 2x = x - 2
\]

\[
d = a_3 - a_2 = (4x - 4) - (3x - 2) = x - 2
\]

The common difference \(d\) is \(x - 2\).

### Step 2: Find the first five terms
The first term \(a_1\) = \(2x\).

The common difference \(d\) is \(x - 2\).

The terms can be expressed as:
- Second term \(a_2 = 2x + (x - 2)\)
- Third term \(a_3 = 2x + 2(x - 2)\)
- Fourth term \(a_4 = 2x + 3(x - 2)\)
- Fifth term \(a_5 = 2x + 4(x - 2)\)

Substituting \(a_1 = 2x\) and \(d = x - 2\):

\[
a_2 = 2x + (x - 2) = 3x - 2
\]
\[
a_3 = 2x + 2(x - 2) = 4x - 4
\]
\[
a_4 = 2x + 3(x - 2) = 5x - 6
\]
\[
a_5 = 2x + 4(x - 2) = 6x - 8
\]

### Step 3: Sum of the first five terms
The sum of the first \(n\) terms of an arithmetic sequence can be calculated using the formula:

\[
S_n = \frac{n}{2} \times (2a_1 + (n-1)d)
\]

For the first five terms (\(n = 5\)):

\[
S_5 = \frac{5}{2} \times (2(2x) + 4(x - 2))
\]

Simplifying inside the parentheses:

\[
S_5 = \frac{5}{2} \times (4x + 4x - 8)
\]

\[
S_5 = \frac{5}{2} \times (8x - 8)
\]

\[
S_5 = \frac{5}{2} \times 8(x - 1)
\]

\[
S_5 = 20(x - 1)
\]

So, the sum of the first five terms is \( \boxed{20(x - 1)} \).

Would you like further details on any of the steps? 

### Related Questions:
1. What is an arithmetic sequence, and how is it defined?
2. How do you determine the common difference in an arithmetic sequence?
3. What is the general formula for the sum of the first \(n\) terms of an arithmetic sequence?
4. How would the sum change if the sequence had a different common difference?
5. How can you derive the sum of an arithmetic sequence from its first and last terms?
6. How do you find a specific term in an arithmetic sequence given the first term and common difference?
7. What are some real-world applications of arithmetic sequences?
8. How does an arithmetic sequence differ from a geometric sequence?

**Tip:** When working with sequences, always check the common difference or ratio first to ensure the sequence is arithmetic or geometric.

Learn how to find the sum of the first five terms of the arithmetic sequence 2x, 3x - 2, 4x - 4. This problem involves determining the common difference and applying the formula for the sum of an arithmetic sequence. Suitable for high school students.

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